1D Kinematics Graphing Free Fall 2016.notebook

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August 30, 2016

www.njctl.org

Algebra Based Physics Kinematics in One Dimension

2016­03­22

www.njctl.org

Jun 29­2:13 PM

Table of Contents: Kinematics• Motion in One Dimension• Average Speed

• Instantaneous Velocity• Acceleration • Kinematics Equation 1

• Kinematics Equation 2• Kinematics Equation 3• Mixed Kinematics Problems

• Average Velocity

• Position and Reference Frame• Displacement

Click on the topic to go to that section

• Free Fall ­ Acceleration Due to Gravity

• Graphing

Jun 29­2:17 PM

Motion in One Dimension

Aug 14­2:24 PM

Distance

We all know what the distance between two objects is...

So what is it? What is distance? What is length?

ALSO ­ you can't use the words "distance" or "length" in your definition; that would be cheating.

Aug 14­2:24 PM

Distance

We'll be using meter as our standard for measuring distance.

The symbol for distance is "d".

And the unit for the meter is "m"

d = 0.2 m

Aug 14­2:24 PM

Time

Similarly, everyone knows what time is...

But try defining it; what is time?

Remember you can't use the word "time" or an equivalent to the word "time", in your definition.

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Aug 14­2:24 PM

Speed

Speed is defined as the distance traveled divided by thetime it took to travel that distance.

speed = distance time

s = d t

speed = meters/sec

Speed is not a fundamental aspect of nature, it is the ratio of two things that are.

Jun 29­2:17 PM

Return toTable ofContents

Average Speed

Sep 8­4:20 PM

Average Speed

The speed we have been calculating is a constant speed over a short period of time. Another name for this is instantaneous speed.

If a trip has multiple parts, each part must be treated separately. In this case, we can calculate the average speed for a total trip.

Determine the average speed by finding the total distance you traveled and dividing that by the total time it took you to travel that distance.

Jul 13­10:15 AM

In physics we use subscripts in order to avoid any confusion with different distances and time intervals.

For example: if an object makes a multiple trip that has three parts we present them as d1, d2, d3 and the corresponding time intervals t1, t2, t3.

Distance and Time Intervals

Jul 13­10:15 AM

The following pattern of steps will help us to find the average speed:

Find the total distance dtotal = d1+ d2+ d3

Find the total time ttotal = t1 + t2 + t3

Use the average speed formula

Average Speed & Non­Uniform Motion

savg = dtotal

ttotal

Sep 8­4:20 PM

Average Speed ­ Example 1

You ride your bike home from school by way of your friend’s house. It takes you 7 minutes (420 s) to travel the 2500 m to his house. You spend 10 minutes there, before traveling 3500 m to your house in 9 minutes (540 s). What was your average speed for this trip?

To keep things clear, we can use a table (graphic

organizer) to keep track of the information...

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Sep 8­4:20 PM

Example 1 ­ Step 1

You ride your bike home from school by way of your friend’s house. It takes you 7 minutes (420 s) to travel the 2500 m to his house. You spend 10 minutes (600 s) there, before traveling 3500 m to your house in 9 minutes (540 s). What was your average speed for this trip?

Segment Distance Time Speed

(m) (s) (m/s)

I

II

III Total /Avg.

Write the given information in the table below:

Sep 8­4:20 PM

Example 1 ­ Step 1

You ride your bike home from school by way of your friend’s house. It takes you 7 minutes (420 s) to travel the 2500 m to his house. You spend 10 minutes (600 s) there, before traveling 3500 m to your house in 9 minutes (540 s). What was your average speed for this trip?

Segment Distance Time Speed

(m) (s) (m/s)

I

II

III Total /Avg.

Write the given information in the table below:

Sep 8­4:20 PM

Example 1 ­ Step 1

You ride your bike home from school by way of your friend’s house. It takes you 7 minutes (420 s) to travel the 2500 m to his house. You spend 10 minutes (600 s) there, before traveling 3500 m to your house in 9 minutes (540 s). What was your average speed for this trip?

Segment Distance Time Speed

(m) (s) (m/s)

I 420II

III Total /Avg.

Write the given information in the table below:

2500

Feb 7­3:58 PM

Example 1 ­ Step 2Next, use the given information to find the total distance and total time

You ride your bike home from school by way of your friend’s house. It takes you 7 minutes (420 s) to travel the 2500 m to his house. You spend 10 minutes (600 s) there, before traveling 3500 m to your house in 9 minutes (540 s). What was your average speed for this trip?

dtotal = d1+ d2+ d3

Segment Distance Time Speed

(m) (s) (m/s)

I 2500 420 II 0 600 III 3500 540

Total /Avg.

Feb 7­4:04 PM

Example 1 ­ Step 2Next, use the given information to find the total distance and total time

You ride your bike home from school by way of your friend’s house. It takes you 7 minutes (420 s) to travel the 2500 m to his house. You spend 10 minutes (600 s) there, before traveling 3500 m to your house in 9 minutes (540 s). What was your average speed for this trip?

ttotal = t1 + t2 + t3

Segment Distance Time Speed

(m) (s) (m/s)

I 2500 420

II 0 600

III 3500 540

Total /Avg. 6000

Feb 7­4:09 PM

Example 1 ­ Step 3Next use total distance and time to find average speed.

You ride your bike home from school by way of your friend’s house. It takes you 7 minutes (420 s) to travel the 2500 m to his house. You spend 10 minutes (600 s) there, before traveling 3500 m to your house in 9 minutes (540 s). What was your average speed for this trip?

savg = dtotal ttotal

Segment Distance Time Speed

(m) (s) (m/s)

I 2500 420

II 0 600

III 3500 540

Total /Avg. 6000 1560

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Sep 8­4:20 PM

Example 1 ­ Solution

Segment Distance Time Speed

(m) (s) (m/s)

I 2500 420

II 0 600

III 3500 540

Total /Avg. 6000 1560 3.85

Next use total distance and time to find average speed.

You ride your bike home from school by way of your friend’s house. It takes you 7 minutes (420 s) to travel the 2500 m to his house. You spend 10 minutes (600 s) there, before traveling 3500 m to your house in 9 minutes (540 s). What was your average speed for this trip?

dtotal ttotal= savg =

6000 m 1560 s =

Jun 29­2:17 PM

Return toTable ofContents

Position and Reference Frames

https://www.njctl.org/video/?v=5mPK2E2GkzA

Aug 14­2:24 PM

Position and Reference Frames

Speed, distance and time didn't require us to define where we started and where we ended up. They just measure how far we traveled and how long it took to travel that far.

However, much of physics is about knowing where something is and how its position changes with time.

To define position we have to use a reference frame.

MOTION IS RELATIVE. THIS MEANS THAT HOW YOU MEASURE SOMEONE ELSE"S MOTION DEPENDS ON WHERE YOU ARE STANDING AND YOUR OWN MOTION AS COMPARED TO SOMETHING OR SOMEONE ELSE

WALKING DEMO

Aug 14­2:24 PM

Position and Reference Frames

A reference frame lets us define where an object is located, relative to other objects.

For instance, we can use a map to compare the location of different cities, or a globe to compare the location of different continents.

However, not every reference frame is appropriate for every problem.

Jun 14­7:42 AM

Who is observing matters!

• Identify the object of interest AND the Observer

• An observer in a spaceship describes the motion of the Sun differently than an observer standing on Earth.

• The observer must be specified!

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

What is motion?

*Motion is a change in an object's position relative to a given observer during a certain change in time.

*Without identifying the observer, it is impossible to say whether the object of interest moved.

*Physicists say motion is relative, meaning that the motion of any object of interest depends on the point of view of the observer.

MOTION IS RELATIVE. THIS MEANS THAT HOW YOU MEASURE SOMEONE ELSE"S MOTION DEPENDS ON WHERE YOU ARE STANDING AND YOUR OWN MOTION AS COMPARED TO SOMETHING OR SOMEONE ELSE

WALKING DEMO

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Jun 14­7:42 AM

Observational experiment

Different observers can describe the same process differently, including whether motion is even occurring.

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Reference frames require:

• An object of reference (or a point on an object if the object is large)

• A coordinate system with a scale for measuring distance

• A clock to measure time

© 2014 Pearson Education, Inc.

Aug 14­2:24 PM

Reference Frame Activity

Aug 14­2:24 PM

In this course, we'll usually:

Define the origin as a location labeled "zero"

Create three perpendicular axes : x, y and z for direction

Use the meter as our unit of measure

Results ­ Reference Frames

Aug 14­2:24 PM

You'll probably find that you need a starting point (an origin)

A set of directions (for instance left­right, forward­backward, up­down)

A unit of measure (how far to go in each direction)

Usually we'll:

Define the origin as a location labeled "zero"

Three axes: x, y and z for direction

The meter as our unit of measure

In this course, we will be solving problems in one­dimension (1D)and two­dimensions (2D). We will start with 1D motion

Typically, we use the x­axis for 1D direction.

+x will usually be to the right

­x would then be to the left

We could define it the opposite way, but unless specified otherwise, this is what we'll assume. We also can think about compass directions in terms of positive and negative. For example, North would be positive and South negative.

The symbol for 1D position is normally "x" or in the case of your book "d" (book uses "d" because it is not associated with the x­y coordinate system).

The Axis

+x­ x

Jun 14­7:42 AM

Linear motion

• Linear motion is a model of motion that assumes that an object, considered as a point­like object, moves along a straight line.

• A car moving along a straight highway can be modeled with linear motion; we simplify the car as a point, which is small compared to the length of the road.

• A tire of the car cannot be modeled with linear motion.

© 2014 Pearson Education, Inc.

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Jun 14­7:42 AM

Motion diagrams contain:

• Dots representing the location of the object for progressive, equal time intervals

• Velocity vectors on each dot representing the velocity of the object at each time interval (the length of the velocity vector represents how fast the object is moving)

• Velocity change arrows showing how the velocity vectors are changing

• The specified location of the observer

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Patterns found from motion diagrams

• The spacing of the dots allows us to visualize motion. • • When the object travels without speeding up or slowing down, the dots are

evenly spaced.

• When the object slows down, the dots get closer together.

• When the object moves faster and faster, the dots get farther apart.

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Velocity change arrows

• Delta means "change in."

• Change always means final (vf) minus initial (vo): what it is now compared to what it was before.

• The arrow above v reminds us that velocity is a vector; it has both magnitude and direction.

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Finding velocity change arrows

• Consider two adjacent velocity vectors, in this example at points 2 and 3.

• Find which vector would need to be added to the velocity corresponding to point 2 to get the velocity corresponding to point 3.

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Constructing a motion diagram

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Quantities for describing motion

• Motion diagrams represent motion qualitatively.

• To analyze situations, we need to describe motion quantitatively.

• These quantities are needed to describe linear motion:

> Time and time interval

> Position, displacement, distance, and path length

> Scalar component of displacement for motion along one axis

© 2014 Pearson Education, Inc.

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Jun 14­7:42 AM

Time and time interval

• The time t is a clock reading.

• The time interval (t2 ­ t1) or Δt is a difference in clock readings. (The symbol delta represents "change in" and is the final value minus the initial value.)

• These are both scalar quantities.

• The SI units for both quantities are seconds (s).

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Position, displacement, distance, and path length

• These quantities describe the location and motion of an object.

• Position is an object's location with respect to the origin (Where you start)

• Displacement is a vector that starts from an object's initial position and ends at its final position (it's the overall change in position).

• Distance is a scalar and represents how much ground an object has covered or how far it has gone during the time period analyzed

• Path length (or total distance traveled) is how far the object moved as it traveled from its initial position to its final position.

> Imagine laying a string along the path the object took. The length of the string is the path length/total distance traveled.

• Run Logger Pro Videos

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Constant Car Lab• Problem: Determine a method to graph motion

• Materials: Meter stick, stop watch, 2 constant velocity cars with different battery combination. VIdeo Camera

• Procedure:> 1. 5 students at uniform intervals from 1 to 3 meters that corresponds to about 1 sec intervals. Start the car.

> 2. When car passes the starting line, shout “GO”. All timers start stop watches. Start the video.

> 3. When car passes a timer at designated time mark the distance (1 sec intervals).

> 4. Record data within the data table determined in class .

> 5. Repeat two more times (only video one of each runs)

> 6. Change out the car and perform steps 1­5 for 2nd car.

.

Jun 14­7:42 AM

Constant Car LabAll Groups

• Make a table for each of your runs. Determine the average time for each distance.• Make a motion diagram on your whiteboard for both cars. Graph 2nd car directly below 1st car. Provide significant figures as appropriate.

YOU MUST USE AVG VALUES. WE ARE NOW IN THE LAB...RAW DATA IS NOT MUCH GOOD BECAUSE OF EXEPRIEMNTAL ERROR THAT AFFECT

PRECISION AND ACURACY.

• Graph both in Logger Pro (remember the video I just played)

Discussion Topics...Be ready To Answer:

• What is different between the two graphs• What is the graph showing.• Can we draw displacement and velocity vectors on the particle motion plot? • Why isn’t the distance between the dots exactly the same between each time interval.• What is the distance travel for both cars.• Lets graph Distance vs Time for both cars (must determine dependant and independent variables). • Determine the slope and tell me what it represents• Can we determine precision and/or accuracy with our experiment. If so make it happen!

Jun 29­2:17 PM

Return toTable ofContents

Displacement

Aug 14­2:24 PM

Displacement

Now that we understand how to define position and distance, we can talk about a change in position; a displacement.

The symbol for "change" is the Greek letter "delta" "Δ".

So "Δx" means the change in x or the change in position

Displacement is a Vector !!!!!!!

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Aug 14­2:24 PM

­x

+y

­y

+x

Displacement

Displacement describes how far you are from where you started, regardless of how you got there.

Aug 14­2:24 PM

­x

+y

­y

+x

Displacement

For instance, if you drive 60 miles from Pennsylvania to New Jersey...

x0(In physics, we label the starting position x0)

Aug 14­2:24 PM

­x

+y

­y

+x

Displacement

and then 20 miles back toward Pennsylvania.

x0 xf

(We also label the final position xf )

Aug 14­2:24 PM

­x

+y

­y

+x

Displacement

You have traveled:

a distance of 80 miles, and

a displacement of 40 miles,

since that is how far you are from where you started

x0 xf

we can calculate displacement with the following formula:Δx = Xf ­ Xo

Aug 14­2:24 PM

Displacement

Distance (the scalar) can only be positive values since it is impossible to travel a negative distance.

Imagine trying to measure a negative length with a meter stick...

Aug 14­2:24 PM

xf xo­x

+y

­y

+xxo xf­x

+y

­y

+x

DisplacementHowever, displacement can be positive or negative since you can end up to the right or left of where you started.

Displacement is positive. Displacement is negative.

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Sep 12­2:26 PM

Vectors and Scalars

Scalar ­ a quantity that has only a magnitude (number or value)

Vector ­ a quantity that has both a magnitude and a direction

Quantity Vector Scalar

Time

Distance

Displacement

Speed

Which of the following are vectors? Scalars?

May 9­9:35 PM

1 How far your ending point is from your starting point is known as:

https://www.njctl.org/video/?v=vxIWZ9_rQFo

A distanceB displacement

C a positive integer

D a negative integer Answ

er

May 9­9:38 PM

2 Starting from the origin, a car travels 4km east and then 7 km west. What is the total distance traveled?

https://www.njctl.org/video/?v=vl2i9fvhOVE

A 3 km B ­3 km C 7 km D 11 km

Answ

er

May 9­9:38 PM

3 Starting from the origin, a car travels 4km east and then 7 km west. What is the net displacement from the original point?

https://www.njctl.org/video/?v=iYNfUacg­9A

A 3 km westB 3 km eastC 7 km west

D 11 km east

Answ

er

Jun 14­7:42 AM

Kinematics Graphs (1D motion/linear motion):

Graph 1: Position Time Graph (P­T Graph)

• Time t is usually the independent variable: the horizontal axis.

• Position x is the dependent variable (position changes with time): the vertical axis (even if the motion is horizontal!).

• A trendline is a best­fit curve that passes as close as possible to the data points.

Jun 14­7:42 AM

Correspondence between a motion diagram and position­versus­time graph

• Kinematics graphs can contain more precise information than motion diagrams.

• The position of each dot on the motion diagram corresponds to a point on the position axis.

• The graph line combines information about the position of an object and the clock reading when this position occurred.

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Jun 14­10:38 AM Jun 14­7:42 AM

Mathematics of linear motion

• A dependent variable, usually y, depends on the value of an independent variable, usually x.

• y(x) = f(x) is an operation that one needs to do if x is an input and y is the output.

• For a straight line, y(x) = kx + b, where k is the slope and b is the y intercept; the value of y when x = 0.

• For motion along the x­axis, we write x(t) to depict that the motion is dependent on time;

• x(t) = kt + b.

Jun 14­7:42 AM

Connecting graphical representations of linear motion to a mathematical representation

• A linear function is written: x(t) = kt + b

• k is the slope of the line; it is the change in the dependent variable divided by the change in the independent variable.

• The slope k can be found from

• k has units of m/s and indicates how the position changes with time.

• k can be positive or negative; it represents not only how fast, but also in which direction an object is moving.

• b is the location of the object at t = 0; it is x0.

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Velocity: Slope of the position­versus­time graph

• If the slope is positive, the object is moving along the +x axis.

• If the slope is negative, the object is moving along the –x axis.

• The magnitude of the slope (which is always positive) is the speed of the object.

• The speed and the direction together are called the velocity of the object.

© 2014 Pearson Education, Inc.

graphing

Return toTable ofContents

Graphing Problems

https://www.njctl.org/video/?v=unLHH0wj01k

Jun 14­10:38 AM

Position‐Time Graphs• Two different runners• At what time do A and B have the same position?• Particle model?

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Jun 14­10:38 AM Jun 14­10:38 AM

Jul 1­8:37 AM

Starting at the position, x0 = 4 m, you travel at a constant velocity of +2 m/s for 6s. a. Determine your position at the times of 0s; 2s; 5s; and 6s.

Answ

er

Jul 1­8:38 AM

Starting at the position, x0 = 4 m, you travel at a constant velocity of +1 m/s for 6s. b. Draw the Position versus Time for your travel during this time.

x(m)

t (s)1

1

2

2 3

3

4 5 6

4

5

6

7

8

9

10

11

Draw a line of best fit to observethe pattern.

Drag and drop the data points on the graphin order to construct the v vs t pattern!

Answ

er

Feb 8­11:01 AM

Starting at the position, x0 = 10 m, you travel at a constant velocity of ­1m/s for 6s. d. Draw the Position versus Time for your travel during this time.

x(m)

t (s)1

1

2

2 3

3

4 5 6

4

5

6

7

8

9

10

11

Draw a line of best fit to observethe pattern.

Drag and drop the data points on the graphin order to construct the v vs t pattern!

Answ

er

Jul 13­2:07 PM

Analyzing Position vs Time Graphs

Recall earlier in this unit that slope was used to describe motion.

The slope in a position vs. time graph is Δx/Δt, which is equal to velocity.

Therefore, slope is equal to velocity on a position vs. time graph.

x(m)

t (s)

Δx

Δt v = Δx/Δt

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Jul 13­2:07 PM

Analyzing Position vs Time GraphsA positive slope is a positive velocity, a negative slope is a negative velocity, and a slope of zero means zero velocity.

x(m)

t (s)

x(m)

t (s)

x(m)

t (s)

positive slopev > 0

negative slopev < 0

zero slopev = 0

A positive velocity means moving in the positive direction, a negative velocity means moving in the negative direction, and zero velocity means not moving at all.

Jun 14­7:42 AM

Equation of motion for constant­velocity linear motion

© 2014 Pearson Education, Inc.

Couple of things:

1. This equation velocity is constant so v is the same throughout the problem. Remember v is a vector (I am using bold instead of a top arrow because smartboard does not have that ability)

2. You have used this equation before in Math: D=Rate*time D=rt

3. Physics: x =velocity*time or vxt (vx is the velocity in the x direction) Note: book sometimes uses d instead of x and therefore does not have a subscript.

Jun 29­2:17 PM

Return toTable ofContents

Average Velocity

Aug 14­2:24 PM

Average Velocity

Speed is defined as the ratio of distance and time.Note Avg Speed and Distance Traveled are SCALERS

Similarly, velocity is defined as the ratio of displacement and timeNOTE: Avg velcoity and displacement are VECTORS!!!

s = dt

ΔxΔtv = Average velocity = time elapsed

displacement

Average speed = distance traveledtime elapsed

Aug 14­2:24 PM

Average VelocitySpeeds are always positive, since speed is the ratio of distance and time; both of which are always positive.

But velocity can be positive or negative, since velocity is the ratio of displacement and time; and displacement can be negative or positive.

s = dt

ΔxΔtv =

Usually, right is positive and left is negative.

Average speed = distance traveledtime elapsed

Average velocity = time elapsed

displacement

Aug 14­2:24 PM

4 You travel 60 meters to the left in 20 s and then you travel 60 meters to the right in 30 s; what is your average velocity?

Answ

er

https://www.njctl.org/video/?v=NQ0nOZFCKZI

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Aug 14­2:24 PM

5 You travel 60 meters to the left in 20 s and then you travel 60 meters to the right in 30 s; what is your average speed?

Answ

er

https://www.njctl.org/video/?v=i_iJgYVaRh8

Jun 29­2:17 PM

Return toTable ofContents

Instantaneous Velocity

https://www.njctl.org/video/?v=3VtT9A5parI

Jul 13­10:58 AM

Instantaneous VelocitySometimes the average velocity is all we need to know

about an object's motion.

For example: A race along a straight line is really a competition to see whose average velocity is the greatest.

The prize goes to the competitor who can cover the displacement in the shortest time interval.

But the average velocity of a moving object can't tell us how fast the object moves at any given

point during the interval Δt.

Aug 21­2:52 PM

Instantaneous Velocity

Average velocity is defined as change in position over time. This tells us the 'average' velocity for a given length or span of time.

Watch what happens when we look for the instantaneous velocity by reducing the amount of time we take to

measure displacement.

Radar Gun Demo!!!!

If we want to know the speed or velocity of an object at a specific point in time (with this radar gun for example), we want to know the instantaneous velocity...

Aug 21­2:52 PM

Instantaneous Velocity

Displacement Time

100m 10 s

Velocity

In an experiment, an object travels at a constant velocity. Find the magnitude of the velocity using the data above.

Aug 21­2:52 PM

Instantaneous Velocity

What happens if we measure the distance traveled in the same experiment for only one second?

What is the velocity?

10 m 1 s

Displacement Time Velocity

100m 10 s 10 m/s

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Aug 21­2:52 PM

Instantaneous Velocity

What happens if we measure the distance traveled in the same experiment for a really small time interval?

What is the velocity?

10 m 1 s 10 m/s

0.001m 0.0001 s

Displacement Time Velocity

100m 10 s 10 m/s

Aug 21­2:52 PM

Displacement Time Velocity

100 m 10 s 10 m/s

10 m 1 s 10 m/s

1.0 m 0.10 s 10 m/s

0.10 m 0.010 s 10 m/s

0.010 m 0.0010 s 10 m/s

0.0010 m 0.00010 s 10 m/s

0.00010 m 0.000010 s 10 m/s

Instantaneous Velocity

Since we need time to measure velocity, we can't know the exact velocity "at" a particular time... but if we imagine a really small value of time and the distance traveled, we can estimate the instantaneous velocity.

Jul 13­11:43 AM

In physics an instant has no duration at all; it refers to a single value of time.

One of the most common examples we can use to understand instantaneous velocity is driving a car and taking a quick look on the speedometer.

Instantaneous Velocity

At this point, we see the instantaneous value of the velocity.

Aug 14­2:24 PM

Instantaneous Velocity

The instantaneous velocity is the same as the magnitude of the average velocity as the time interval becomes very very short.

ΔxΔt as Δt 0

v =

Jun 14­10:38 AM

Average vs. Instantaneous?

Jun 14­7:42 AM

Graphing Velocity

Graph 2: The V­T Kinematics Graph

© 2014 Pearson Education, Inc.

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Jun 29­2:01 PM

v(m/s)

t (s)

The graph below shows velocity versus time.

How do you know the velocity is constant?

Velocity Graphing Activity

Answ

erJun 29­2:01 PM

v(m/s)

t (s)

The graph below shows velocity versus time.

When is the velocity increasing? Decreasing? Constant?Discuss.

Velocity Graphing Activity

Jun 29­2:03 PM

Use the graph to determine the Average Velocity of (a).

Velocity Graphing Activity

Answ

er

v(m/s)

t (s)

1

3

2

4

2 4 6

a.)

Jun 29­2:03 PM

b.)

11

3

2

2 4 6

4v

(m/s)

t (s)

Use the graph to determine the Average Velocity of (b)

from 0s to 1s

from 1s to 3s

from 3s to 4s

from 4s to 5s

from 3s to 5s

Velocity Graphing Activity

Answ

er

Jun 29­2:03 PM

v(m/s)

t (s)

v(m/s)

t (s)

a.)

b.)

Use the graph to determine the Instantaneous Velocity of (a) at 2 seconds 1

3

2

4

2 4 6

11

3

2

2 4 6

4

Velocity Graphing Activity

Answ

er

Jun 29­2:03 PM

v(m/s)

t (s)

v(m/s)

t (s)

a.)

b.)Use the graph to determine the Instantaneous Velocity of (b) at 2 seconds

1

3

2

4

2 4 6

11

3

2

2 4 6

4

Velocity Graphing Activity

Answ

er

1D Kinematics Graphing Free Fall 2016.notebook

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August 30, 2016

Aug 14­2:24 PM

Instantaneous Velocity

(a) When the velocity of a moving object is a constant the instantaneous velocity is the same as the average.

v(m/s)

t (s)

v(m/s)

t (s)

These graphs show (a) constant velocity and (b) varying velocity.

(b) When the velocity of a moving object changes its instantaneous velocity is different from the average velocity.

Jul 1­8:38 AM

Starting at the position, x0 = 4 m, you travel at a constant velocity of +2 m/s for 6s. c. Draw the Velocity versus Time graph for your trip.

v(m/s)

t (s)1

1

2

2 3

3

4 5 6

4

Drag and drop the data points on the graphin order to construct the v vs t pattern!

Draw a line of best fit to observethe pattern.

Answ

er

Jun 14­7:42 AM

Finding displacement from a velocity graph

• Displacement x – x0 between t0 = 0 and time t is the area under the V­T graph.

• Remember v=

• So looking at the V­T graph:

> Area is width times height = vx(t–t0)

• Since vx= (x – x0)/(t–t0), (x – x0) = vx(t–t0)

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Finding displacement from a velocity graph

© 2014 Pearson Education, Inc.

­ Displacement can be negative

­ Area under the curve IS NOT Path traveled or total distance!!!!

Jun 14­7:42 AM

When velocity is not constant

• On a velocity­versus­time graph, velocity will be a straight line only if it is constant.

• Instantaneous velocity is the velocity of an object at a particular time.

• Average velocity is the ratio of the change in position and the time interval during which this change occurred.

• For motion at constant velocity, the instantaneous and average velocities are equal; for motion with changing velocity, they are not.

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Acceleration characterizes the rate at which the velocity of an object is changing

• The simplest type of linear motion with changing velocity occurs when the velocity of the object increases or decreases by the same amount during the same time interval (a constant rate of change).

• In this simple case, the velocity­versus­time graph will be linear.

© 2014 Pearson Education, Inc.

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August 30, 2016

Jun 14­7:42 AM

Finding acceleration from a velocity­versus­time graph

• Acceleration is the slope of the velocity­versus­time graph:

• A larger slope indicates the velocity is increasing at a faster rate.

• Velocity is a vector quantity; therefore acceleration is also a vector quantity.

• The average acceleration of an object during a time interval is

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Acceleration

© 2014 Pearson Education, Inc.

Jun 29­2:17 PM

Return toTable ofContents

Acceleration

https://www.njctl.org/video/?v=jGbVA3e9Op4

Jun 15­9:43 AM

Acceleration

• You can feel a difference between uniform (constant) and nonuniform (accelerated) motion.

• When you move in a nonuniform motion, you feel pushed or pulled.

• In contrast, when you are in uniform motion and your eyes are closed, you feel as though you are not moving at all.

Jun 15­9:43 AM

Acceleration• So far, we’ve talked about “how far” and “how fast”…now

we want to study “how fast an object is getting faster”• The rate at which an object’s velocity changes is called

acceleration.• Acceleration is measured in m/s/s or m/s2.• What is meant by “per”

Jun 15­9:43 AM

Displaying Acceleration on a Motion Diagram

• To determine the length and direction of an average acceleration vector,

• subtract two consecutive velocity vectors, as shown below.

1D Kinematics Graphing Free Fall 2016.notebook

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August 30, 2016

Jun 15­9:43 AM

Displaying Acceleration on a Motion Diagram

• Direction: You will have: ∆v = vf ­ vi = vf + (­vi)

• Acceleration Vector Always the same as ∆

• Magnitude: Then divide by the time interval, ∆t.

• Remember:

Jun 15­9:43 AM

Finding the Acceleration Vector

Text: p. 69

Jun 15­9:43 AM

Position, Velocity, Acceleration

Note: Constant Velocity =No Change in velocity

ACCELERATION = 0

Jun 15­9:43 AM

Positive and Negative Acceleration

• When the object’s acceleration is in the same direction as its velocity,

• the object’s velocity increases.

• When they are in opposite directions,• the velocity decreases.• So……..…

Jun 15­9:43 AM

Positive and Negative Acceleration

Jun 15­9:43 AM

Velocity and Acceleration

1D Kinematics Graphing Free Fall 2016.notebook

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August 30, 2016

Jun 15­9:43 AM

3 Ways to Show Acceleration

Aug 14­2:24 PM

Acceleration

Acceleration is the rate of change of velocity.

a = ΔvΔt

acceleration = change of velocityelapsed time

Aug 14­2:24 PM

Acceleration

Acceleration is a vector, although in one­dimensional motion we only need the sign.

Since only constant acceleration will be considered in this course, there is no need to differentiate between average and instantaneous acceleration.

Aug 14­2:24 PM

Units for Acceleration

= m/ss m/s2a = Δv

Δt

Jun 14­7:42 AM

When is acceleration negative?

• Acceleration can be positive or negative.

• If an object moving in the positive direction is slowing down, its velocity­versus­time graph has a negative slope, corresponding to a decreasing speed and negative acceleration.

• An object can have a negative acceleration and speed up!

• Consider an object moving in the negative direction with a negative component of velocity, but whose speed is increasing in magnitude.

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Determining the velocity change from the acceleration

From the slope of the velocity­versus­time graphs, we see:

• ax = (vx − v0x)/(t−t0), therefore: ax (t−t0) = (vx − v0x)

• Set t0 = 0 to simplify (the clock starts at 0) and move v0x to the other side:

• vx = v0x + axt

© 2014 Pearson Education, Inc.

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August 30, 2016

Jun 14­7:42 AM

Position as a function of time

• The equation for displacement can be found from the area between the velocity­versus­time graph line and the time axis.

© 2014 Pearson Education, Inc.

Aug 14­2:24 PM

Motion at Constant AccelerationGraphical Approach

v(m/s)

t (s)

A = lw

If the area under the graph is length x width (A = lw), then:

A = v0t Since we know that v = ,then area is really Δx.

A = Δx = v0t

Δx t

Aug 14­2:24 PM

Motion at Constant AccelerationGraphical Approach

v(m/s)

t (s)

A = ½bh

If the area under this graph is ½ base x height, then:

A = ½tΔv Since we know that a = ,Δv = at.

A = Δx = ½t(at) = ½at2

Δv t

Aug 14­2:24 PM

Motion at Constant AccelerationGraphical Approach

v(m/s)

t (s)

Therefore, the area under a velocity vs. time graph is displacement. It can be calculated by combining the previous two results.

A = Δx = v0t + ½at2

x ­ x0 = v0t + ½at2

x = x0 + v0t + ½at2

½at2

v0t

Jun 14­7:42 AM

Position of an object during linear motion with constant acceleration

© 2014 Pearson Education, Inc.

Jun 14­7:42 AM

Graph of position versus time for constant acceleration motion

• Position is quadratic in time (there is a t2 term), so the graph is parabolic.

• The slopes of the tangent lines (indicating the instantaneous velocity) are different for different times.

© 2014 Pearson Education, Inc.

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21

August 30, 2016

Jul 30­11:42 AM

Graphing Acceleration

Graph 3: Accleration vs Time Graph

Board Demos: Putting All The Graphs Together!!!!

PLEASE NOTE: In this class acceleration will be 0 or constant!!! You need calculus to deal with an acceleration that changes with time.

Jun 15­9:43 AM

Question 1

• Which of the following statements correctly define acceleration?

• Acceleration is the rate of change of displacement of an object.

• Acceleration is the rate of change of velocity of an object.• Acceleration is the amount of distance covered in unit

time.• Acceleration is the rate of change of velocity of an object.

Jun 15­9:43 AM

Question 2

• What happens when the velocity vector and the acceleration vector of an object in motion are in same direction?

• A. The acceleration of the object increases.• B. The velocity of the object increases.• C. The object comes to rest.• D. The velocity of the object decreases.

May 9­9:55 PM

6 A horse gallops with a constant acceleration of 3m/s2 . Which statement below is true?

A The horse's velocity doesn't change.

B The horse moves 3m every second.

C The horse's velocity increases 3m every second.

D The horse's velocity increases 3m/s every second.

Answ

er

Question 3

Jun 14­7:42 AM

The Kinematic Equations of Motion or The FAB FOUR (acceleration is constant)

(Plug 2 into 3 get 4)

Aug 14­2:24 PM

Solving Problems

After you read the problem carefully:

1. Draw a diagram (include coordinate axes).

2. List the given information.

3. Identify the unknown (what is the question asking?)

4. Choose a formula (or formulas to combine)

5. Rearrange the equations to isolate the unknown variable.

6. Substitute the values and solve!

7. Check your work. (You can do the same operations to the units to check your work ... try it!)

1D Kinematics Graphing Free Fall 2016.notebook

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August 30, 2016

Jun 29­2:17 PM

Return toTable ofContents

Kinematics Equation 1

Δx t= (v + vo)

2

Aug 14­2:24 PM

Motion at Constant AccelerationIf velocity is changing at a constant rate, the average velocity is just the average of the initial and final velocities.

And we learned earlier that Δxtv =

v = v + vo2

Some problems can be solved most easily by using these two equations together.

Δxt = v + vo

2

Δx t= (v + vo)

2

https://www.njctl.org/video/?v=mQ8GOB­nL3c

Sep 6­9:59 PM

7 Starting from rest you accelerate to 20 m/s in 4.0s. What is your average velocity?

Answ

er

https://www.njctl.org/video/?v=RbtF1rQASGw

Jun 29­2:17 PM

Return toTable ofContents

Kinematics Equation 2

v = vo + at

Aug 21­4:30 PM

a = ΔvΔt

Motion at Constant Accelerationbut since "Δ" means change

Δv = v ­ vo and

Δt = t ­ to if we always let to = 0, Δt = t

Solving for "v"

This equation tells us how an object's velocity changes as a function of time.

a = v ­ vot

at = v ­ vov ­ vo = at

v = vo + at

Aug 14­2:24 PM

8 Starting from rest, you accelerate at 8.0 m/s2 for 9.0s. What is your final velocity?

Answ

er

https://www.njctl.org/video/?v=iyPkgH3fJ­0

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August 30, 2016

Aug 14­2:24 PM

https://www.njctl.org/video/?v=X_jWqMS3myk

9 You have an initial velocity of 5.0 m/s. You then experience an acceleration of ­1.5 m/s2 for 4.0s; what is your final velocity?

Answ

erJul 13­2:07 PM

Motion at Constant Acceleration

(slope)m =Δy/Δx

(slope of velocity vs. time)a =Δv/Δt

The graph on the right has a slope of Δv/Δt, which is equal to acceleration. Therefore, the slope of a velocity vs. time graph is equal to acceleration.

Jul 13­3:49 PM

10 The velocity as a function of time is presented by the graph. What is the acceleration?

Answ

er

Jul 14­9:59 AM

11 The velocity as a function of time is presented by the graph. Find the acceleration.

Answ

er

https://www.njctl.org/video/?v=owkR­TAxlgs

Jul 14­10:07 AM

The acceleration graph as a function of time can be used to find the velocity of a moving object. When the acceleration is constant the velocity is changing by the same amount each second. This can be shown on the graph as a straight horizontal line.

Motion at Constant Acceleration

In order to find the change in velocity for a certain limit of time we need to calculate the area under the acceleration line that is limited by the time interval.

Jul 14­10:07 AM

Motion at Constant Acceleration

The change in velocity during first 12 seconds is equivalent to the shadowed area

(4m x 12s = 48m).

The change in velocity during first 12 seconds is 48 m/s.

s2 s

1D Kinematics Graphing Free Fall 2016.notebook

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August 30, 2016

Jul 14­10:33 AM

12 The following graph shows acceleration as a function of time of a moving object. What is the change in velocity during first 10 seconds?

Answ

er

https://www.njctl.org/video/?v=D3bQyx0ygFM

Jun 29­2:17 PM

Return toTable ofContents

Kinematics Equation 3

x = xo + vot + ½at2

Aug 14­2:24 PM

Motion at Constant Acceleration

We can combine these three equations to derive an equation which will directly tell us the position of an object as a function of time.

Δxtv = v = v + vo2

Δx tv =

x ­ xo = ½ (v + vo)t x ­ xo = ½vt + ½vot x = xo + ½vot + ½vt x = xo + ½vot + ½(vo + at)t x = xo + ½vot + ½vot + ½at2

x = xo + vot + ½at2

v = vo + at

Sep 3­8:03 PM

13 An airplane starts from rest and accelerates at a constant rate of 3.0 m/s2 for 30.0 s before leaving the ground. How far did it move along the runway?

Answ

er

https://www.njctl.org/video/?v=OhB6n5VNJpA

Sep 3­8:03 PM

14 A Volkswagen Beetle moves at an initial velocity of 12 m/s. It coasts up a hill with a constant acceleration of –1.6 m/s2 . How far has it traveled after 6.0 seconds?

Answ

er

https://www.njctl.org/video/?v=Iv8l_­RG_fY

Sep 3­8:09 PM

15 A train pulling out of Grand Central Station accelerates from rest at a constant rate. It covers 800 meters in 20 seconds. What is its rate of acceleration?

Answ

er

https://www.njctl.org/video/?v=A2YjfXJW1pA

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August 30, 2016

Sep 3­8:12 PM

16 A Greyhound bus traveling at a constant velocity starts to accelerate at a constant 2.0 m/s2 . If the bus travels 500 meters in 20 seconds, what was its initial velocity?

Answ

er

https://www.njctl.org/video/?v=MQxR6xG­AiM

Jun 29­2:17 PM

Return toTable ofContents

Kinematics Equation 4

Aug 14­2:24 PM

Motion at Constant Acceleration

We can also combine these equations so as to eliminate t:

(v+vo)/2

Sep 6­10:41 PM

17 You accelerate from 20m/s to 60m/s while traveling a distance of 200m; what was your acceleration?

Answ

er

v2 = vo2 + 2aΔxa = (v2 ­ v02)/2Δxa = ((60m/s)2 ­ (20m/s)2)/2(200m)a =8 m/s2

Sep 6­10:47 PM

18 Beginning with a velocity of 25m/s, you accelerate at a rate of 2.0m/s2 . During that acceleration you travel 200m; what is your final velocity?

Answ

er

Sep 6­10:41 PM

19 A ball with an initial velocity of 25m/s is subject to an acceleration of ­9.8 m/s2 ; how high does it go before coming to a momentary stop?

Answ

er

https://www.njctl.org/video/?v=ReTH6IKoUYs

1D Kinematics Graphing Free Fall 2016.notebook

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August 30, 2016

Mixed

Return toTable ofContents

Mixed Kinematics Problems

Feb 7­7:01 PM

20 An arrow is projected by a bow vertically up with a velocity of 40 m/s, and reaches a target in 3 s. How high is the target located?

Answ

er

x = xo + vot + ½at2x = vot + ½at2x = (40m/s)3s + ½(­9.8m/s2)(3s)2

x = 120m + (­44.1m)

x = 75.9m

Feb 7­6:59 PM

21 An object accelerates from rest, with a constant acceleration of 8.4 m/s2, what will its velocity be after 11s?

Answ

er

Feb 7­6:57 PM

22 An object accelerates from rest to a velocity of 34 m/s over a distance of 70 m. What was its acceleration?

Answ

er

Sep 6­10:41 PM

a. Describe, in words, the velocity of each of the cars. Make sure you discuss each car’s speed and direction.

Position (m)

Time (s)

The position versus time graph, below, describes the motion of three different cars moving along the x­axis.

Answ

er

Sep 6­10:41 PM

b. Calculate the velocity of each of the cars.

The position versus time graph, below, describes the motion of three different cars moving along the x­axis.

Position (m)

Time (s) Answ

er

1D Kinematics Graphing Free Fall 2016.notebook

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August 30, 2016

Jul 1­8:53 AM

v(m/s)

t (s)

c. Draw, on one set of axes, the Velocity versus Time graph for each of the three cars.

Position (m)

Time (s)

Answ

erFeb 7­7:29 PM

v(m/s)

t (s)

11

3

2

2 4 6

4

The velocity vs time graph, below, describes the motion

of an object moving along the x­axis.

Answ

er

When is velocity zero?

Feb 6­7:01 PM

v(m/s)

t (s)

11

3

2

2 4 6

4

The velocity vs time graph, below, describes the motion

of an object moving along the x­axis.

Describe in words what is happening to the speedduring the following intervals.a) 0s to 1s b) 1s to 3s c) 3s to 4 secd) 4s to 5s e) 5s to 6s

Answ

er

Jun 8­4:54 PM

23 The velocity vs time graph, below, describes the motion of an object moving along the x­axis.

v(m/s)

t (s)

11

3

2

2 4 6

4

Determine the average speed during the following intervals.a) 0s to 1s b) 1s to 3s c) 3s to 4 secd) 4s to 5s e) 5s to 6s f) 3s to 5s

Feb 6­7:01 PM

v(m/s)

t (s)

11

3

2

2 4 6

4

The velocity vs time graph, below, describes the motion

of an object moving along the x­axis.

Determine the displacement during the following intervals.a) 0s to 1s b) 1s to 3s c) 3s to 4 secd) 4s to 5s e) 5s to 6s

a) 0s to 1s Vavg = +2m/s

b) 1s to 3s Vavg = +4m/s

c) 3s to 4s Vavg = +2m/s

d) 4s to 5s Vavg = ­2m/s

e) 5s to 6s Vavg = ­4m/s

f) 3s to 5s Vavg = 0m/s

Vavg = (Vf + Vi)/2

Answ

er

Feb 7­7:51 PM

24 Determine the net displacement during the first four seconds of travel.

v(m/s)

t (s)

11

3

2

2 4 6

4

The velocity vs time graph, below, describes the motion

of an object moving along the x­axis.

Answ

er

1D Kinematics Graphing Free Fall 2016.notebook

28

August 30, 2016

Jun 15­9:43 AM

Determining Acceleration from av‐t Graph

• Graphs A, B, C, D, and E, as shown on the right, represent the motions of five different runners.

• Assume that the positive direction has been chosen to be east.

• What are the accelerations for each.

Jun 15­9:43 AM

Determining Acceleration from av‐t Graph

• The slopes of Graphs A and E are zero. Thus, the accelerations are zero. Both Graphs A and E show motion at a constant velocity— Graph A to the east and Graph E to the west.

• Graph B shows motion with a positive velocity. The slope of this graph indicates a constant, positive acceleration.

Jun 15­9:43 AM

Determining Acceleration from av‐t Graph

• Graph C has a negative slope, showing motion that begins with a positive velocity, slows down, and then stops. This means that the acceleration and velocity are in opposite directions.

• The point at which Graphs C and B cross shows that the runners’ velocities are equal at that point. It does not, however, give any information about the runners’ positions..

Jun 15­9:43 AM

Determining Acceleration from av‐t Graph

• Graph D indicates movement that starts out toward the west, slows down, and for an instant gets to zero velocity, and then moves east with increasing speed.

Jun 15­9:43 AM

Determining Acceleration from av‐t Graph

• The slope of Graph D is positive.• Because the velocity and

acceleration are in opposite directions, the velocity decreases and equals zero at the time the graph crosses the axis.

• After that time, the velocity and acceleration are in the same direction and the velocity increases.

Aug 14­2:24 PM

Summary

• Kinematics is the description of how objects move with respect to a defined reference frame.

• Displacement is the change in position of an object.

• Average speed is the distance traveled divided by the time it took; average velocity is the displacement divided by the time.

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August 30, 2016

Feb 7­11:32 AM

• Instantaneous velocity is the limit as the time becomes infinitesimally short.

• Average acceleration is the change in velocity divided by the time.

• Instantaneous acceleration is the limit as the time interval becomes infinitesimally small.

Summary (continued)

Aug 14­2:24 PM

Summary (continued)

• There are four equations of motion for constant acceleration, each requires a different set of quantities.

v2 = vo2 + 2a(x ­ xo)

x = xo + vot + ½at2v = v o + at

v = v + vo2

Jun 15­9:43 AM

Describing Motion with Graphs

• Plot and interpret a distance­time graph and a velocity­time graph. 2. Deduce from the shape of a distance­time graph when a body is:

(a) at rest (b) moving with uniform velocity(c) moving with non­uniform velocity

3. Deduce from the shape of a velocity­time graph when a body is: (a) at rest (b) moving with uniform velocity(c) moving with uniform acceleration (d) moving with non­uniform acceleration

4. Calculate the area under a velocity­time graph to determine the distance travelled for motion with uniform velocity or uniform acceleration.

Jun 15­9:43 AM

Key Concepts

Position­time Graph1. Slope of the Position­time Graph is the velocity of the moving

object

Velocity­time Graph• Slope of the Velocity­time Graph is the acceleration of the moving

object.

2. Area under the Velocity­time Graph is the distance travelled.

Jun 29­2:17 PM

Return toTable ofContents

Free Fall: Acceleration Due to GravityOr Gravity is Making me Fall

Jun 14­7:42 AM

Free fall

• The hypothetical motion of falling objects in the absence of air is a model of the real process and is called free fall.

• Galileo hypothesized, based on experiments, that free fall occurs exactly the same way for all objects regardless of their mass and shape.

• Galileo hypothesized that speed increases in the simplest way—linearly with time of flight; the acceleration of free­falling objects is constant.

1D Kinematics Graphing Free Fall 2016.notebook

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August 30, 2016

Jun 15­10:18 AM

• Galileo concluded:

• neglecting the effect of the air, all objects in free fall had the same acceleration.

• It didn‛t matter what they were made of, how much they weighed, what height they were dropped from, or whether they were dropped or thrown.

• The acceleration of falling objects, given a special symbol, g, is equal to 9.80 m/s2. We can use 10 m/s2 in most of our calculations.

• The acceleration due to gravity is the acceleration of an object in free fall.

Acceleration Due to Gravity

Jun 15­10:18 AM

Acceleration Due to Gravity

• So……………………………………..

• Object is in Free Fall when:

> The only force acting on it is the force of gravity

> Its not touching other objects

> There is no air resistance (The vacuum that you can breath)

• When on earth an object is in Free Fall

> ay= ­g =­9.81m/s2 (or 10 m/s2 for calculations)

> Not on earth little g is different!!!!!

• gmercury= 3.61m/s2 gmoon = 1.62 m/s2 (approx 1/6 of earth)• gvenus= 8.83 m/s2 gmars = 3.75m/s2

Jun 15­10:15 AM

Acceleration Due to Gravity

• ­g because acceleration is down….its in the negative y direction

• However…acceleration due to gravity (g) is positive!!!! Its only when we talk about it in an xy system we say negative

• g varies from place to place on earth…However, for our problems we are going to assume a constant g so we can use our Uniform Accelerated Motion equations (kinematic equation).

• Pick a g (most cases 9.81/10.0 m/s2) and stick with it (Global vs local...specific location)

• Little g is actually not called “gravity.” g is the free­fall acceleration.

Jun 15­10:15 AM

If everything accelerates at the same rate, does that mean everything falls at the same rate?

• Even they have a different weight? Yes

• Even if they are different sizes and shapes? Yes

• However, remember its YES as long as we do not include air resistance ( for example a feather falls at the same rate but is very affected by air resistance where the lacrosse ball is affected very little.

Jun 14­7:42 AM

Motion diagrams and graphs for free fall

© 2014 Pearson Education, Inc.

Sep 15­10:22 AM

Free Fall

All unsupported objects fall towards Earth with the same acceleration. We call this acceleration the "acceleration due to gravity" and it is denoted by g.

g = 9.8 m/s2

Keep in mind, ALL objects accelerate towards the earth at the same rate.

g is a constant!

Click here to watch Galileo's famousexperiment performed on the moon

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August 30, 2016

Sep 22­1:01 PM

It speeds up(negative acceleration)g = ­9.8 m/s2

It stops momentarily.v = 0g = ­9.8 m/s2

It slows down.(negative acceleration)g = ­9.8 m/s2What happens when it goes up?

What happens when it goes down?

What happens at the top?

It returns with itsoriginal velocity.What happens when it lands?

Free Fall

An object is thrown upward with initial velocity, vo

(Click on question for answer.)

Jun 9­11:13 PM

It speeds up.(negative acceleration)g = ­9.8 m/s2

It stops momentarily.v = 0g = ­9.8 m/s2

An object is thrown upward with initial velocity, vo

It slows down.(negative acceleration)

g = ­9.8 m/s2

It returns with itsoriginal velocity.

Free Fall Answers

Jun 30­11:38 AM

a

v0

On the way up:

a

v1

v1av2

v2

a

a

va

av0

On the way down:

v1

v1v2

v2

v

vt = 0 s

t = 1 s

t = 2 s

t = 3 s t = 0 s

t = 1 s

t = 2 s

t = 3 s

Free Fall

Jun 30­12:07 PM

v(m/s)

t (s)

An object is thrown upward with initial velocity, vo

It stops momentarily.v = 0g = ­9.8 m/s2

It returns with itsoriginal velocity but in the opposite direction.

For any object thrown straight up into the air, this is what the velocity vs. time graph looks like.

Free Fall

Jun 15­10:15 AM

Free Fall

• The figure shows the motion diagram for an object that was released from rest and falls freely. The diagram and the graph would be the same for all falling objects.

Jun 15­10:15 AM

Question 1

• A ball is tossed straight up in the air. At its very highest point, the ball’s instantaneous acceleration ay is

• Positive.• Negative.• Zero.

© 2015 Pearson Education, Inc.

1D Kinematics Graphing Free Fall 2016.notebook

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August 30, 2016

Jun 15­10:15 AM

Question 2• An arrow is launched vertically upward. It

moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. At which point of the trajectory is the arrow’s acceleration the greatest? The least? Ignore air resistance; the only force acting is gravity.

© 2015 Pearson Education, Inc.

Jun 15­10:15 AM

Question 3• An arrow is launched vertically upward. It moves straight up to a

maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity.

Jun 15­10:15 AM

Question 3• An arrow is launched vertically upward. It moves straight up to a

maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity.

D

Jun 15­10:15 AM

Question 4: Analyzing a rock’s fall

A heavy rock is dropped from rest at the top of a cliff and falls 100 m before hitting the ground. How long does the rock take to fall to the ground, and what is its velocity when it hits? Draw it……..

yi = 100 m.

Jun 15­10:15 AM

Question 4: Analyzing a rock’s fall (cont.)

• Free fall is motion with constant acceleration: • ay = g.. Using (vy)i = 0 m/s and ti = 0 s, we find

• We can now solve for tf:

• Now that we know the fall time, we can use the first kinematic equation to find (vy)f:

May 9­10:00 PM

25 A ball is dropped from rest and falls (do not consider air resistance). Which is true about its motion?

https://www.njctl.org/video/?v=lqrx6I6fFPo

A acceleration is constant

B velocity is constant

C velocity is decreasing

D acceleration is decreasing Answ

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1D Kinematics Graphing Free Fall 2016.notebook

33

August 30, 2016

Sep 15­1:38 PM

26 A ball is thrown down off a bridge with a velocity of 5 m/s. What is its velocity 2 seconds later?

Answ

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Sep 15­1:42 PM

27 An arrow is fired into the air and it reaches its highest point 3 seconds later. What was its velocity when it was fired?

Answ

er

Sep 15­1:44 PM

28 A rocket is fired straight up from the ground. It returns to the ground 10 seconds later. What was its launch speed?

Answ

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