Chapter 2. Concepts of Motion

Kinematics – the mathematical description and analysis of motion.

Chapter 2Student Learning Objectives

• To differentiate between concepts of position, velocity, and acceleration.

• To gain experience with graphical addition and subtraction of vectors.

• To understand why the sign of the acceleration vector does not indicate speeding up or slowing down.

• To begin the process of learning to analyze problem statements and to translate the information into other representations.

• To make and interpret graphs of motion.

The position vector

• The position vector shows the distance (magnitude) and direction of an object from an arbitrary origin. The SI unit is the meter (m).

m 0.2x m 0.5x

Examples of one-dimensional position vectors

0

For 1-D vectors parallel to a coordinate axis, the vector is equivalent to its scalar component; therefore direction can be specified by positive and negative signs.In this class, negative signs attached to vector (component) quantities mean down, left, west or south.Negative signs do not mean going backward, slowing down, or less than positive quantities of the same magnitude when used with a vector quantity.

m 0.2x

One-dimensional position vectors

• The position vector shows where an object is, relative to the origin, but not in which direction the object is going.• The numbers show the object is moving to the right, but

the first two position vectors point to the left.• The object started at the position x = -2m and moved right

• • • •t=0s t=1s t=2s t=3s

x origin

position initial ox position final x

ntdisplaceme oxxx

The displacement vector represents the change in position of an object relative to an origin. It has a magnitude equal to the shortest distance between the two positions.The symbol Δ (delta) always means final value minus initial value.

The displacement vector

position initial ox position final x

ntdisplaceme oxxx

The displacement vector

x

x ox

position initial ox position final x

ntdisplaceme oxxx

Unlike the position vector, the displacement vector always points in the direction of motion.Obtaining the displacement vector is an example of graphical vector subtraction.

The displacement vector

m 0.2ox

m 0.7x

?x

Displacement Vector 1

Which way does the displacement vector point?

m 0.2x

m 0.7ox

?x

Displacement Vector 2

Which way does the displacement vector point?

m 0.2ox

?x

m 0.5xDisplacement Vector 3

Which way does the displacement vector point?

DisplacementThree runners start at the same place. Shaun runs 4.0 km due east and then runs 1.0 km due west. Mark runs 3.0 km due east. Jeff runs 2.0 km due west and then runs 5.0 km due east. Which of the following is true concerning the displacement of each runner?

A. Mark's displacement equals Jeff's displacement, but Shaun's displacement is different. B. Shaun's displacement equals Mark's displacement, but Jeff's displacement is different.     C. Shaun, Mark, and Jeff have the same displacements. D. Shaun's displacement equals Jeff's displacement, but Mark's displacement is different. E. Shaun, Mark, and Jeff have different displacements.

Average velocity is the rate of change of position during a time interval.Since change of position (Δx) is displacement, the average velocity can be determined by dividing dispacement by the time interval.

timeElapsedntDisplaceme velocityAverage

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xxxv

SI unit for velocity: meters per second (m/s)

2.2 Speed and Velocity

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The direction of the velocity vector is:• the same as that of the displacement vector

(note displacement vector divided by a scalar)• always in the direction of motion• for one-dimensional motion, we use positive

and negative to determine direction, since we are really using components.

Graph of a Straight Line• The vertical axis value comes first

when naming graphs• The standard format is:

vertical axis = m(horizontal axis) + b, where m is the slope and b is the y-

intercept.• The normal variable for horizontal

position is x and the quantity on the horizontal axis is time (t) so the equation for this line is:

x = mt + b

Position as a function of time

Graph of a Straight Linem = “rise over run”, in this

case Δx/Δt, where:Δx = x2-x1 and Δt = t2-t1 for

any two points on the line.

Note that the slope has units, in this case:

Δx/Δt = 8m/2s = +4 m/s.In this case the slope has

units of velocity and represents the velocity of the object being graphed.

Position as a function of time

Graphical Analysis of Contstant Velocity

If an object moved 4 m every second, a graph of its motion a graph of its motion would be a straight line. What is the equation of this line, using the appropriate variables?

A. y = 4tB. y = 4xC. x = 4tD. t = 4x

Position as a function of time

Graphical Analysis of Constant Velocity

The equation of this line is:x = 4t + 0

The general format for uniform motion is:

x = vΔt + x0 wherev= constant velocity, m/sx0= initial position (position at the start of the time interval, usually t=0) One could rewrite the equation:

The slope of a linear position graph is the velocity.

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o

xxxv

The slope of the position graph tells us the rate of change of position during the time interval

Position as a function of time

Equation of a lineFind the equation of a line

if you know the x,t coordinates of 2 points:

• use Δx/Δt to find m• substitute the values of

one of the points into the equation to find b:

x1 = mt1 + bb = x1 – mt1

orb = x2 – mt2

Position as a function of time

At the end of 10 seconds, the object is 5 meters away from the origin. What is the equation of the line?

A. x = 0.5tB. x = 2t +2C. x = 0.5 t + 2D. x = 0.3 t + 2

Instantaneous velocity

When the object is speeding up or slowing down, the position graph will be curved (blue). By drawing a straight line from beginning to some known end (green), we can determine the average velocity:

But that doesn’t tell us how fast the object was going at any particular time.

sm

t 2050

xv

Instantaneous velocity

The slope of the tangent line drawn to the curve at a given time (t = 20 s) tells us the instantaneous velocity at t = 20 s. Pick any 2nd point on the tangent line to find the slope (46,20), (72,25).

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tvinst /2.5

526

x

2.3 Acceleration

If an object speeds up OR slows down during some time interval, it has an acceleration.Acceleration is a vector that represents the rate of change of velocity for a given time interval.

Often the term “deceleration” is used for objects slowing down.However, it is absolutely correct to say that an object that is slowing down is accelerating.

2.3 Acceleration

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vvva

• SI unit for acceleration: meters per second per second (m/s/s or m/s2).• The acceleration vector points in the same direction

as the Δv vector. • The a vector does NOT ALWAYS point in the same

direction as either v0 or v

Direction of Acceleration- it’s probably not what you think

• Acceleration is a vector quantity. Recall our previous assertion regarding the direction of vectors:

• “Negative signs attached to vector quantities mean down, left, west, or south. They do not mean going backward, slowing down, or less than positive quantities….

• So what is “negative acceleration” if not slowing down?

Direction of Acceleration – Here’s the rule• If the object is speeding up, the

velocity vectors and the acceleration vectors point in the same direction, whether that be positive or negative.

• If the object is slowing down, the velocity vectors and acceleration vectors point in opposite directions.

• The direction of the velocity vector is always the direction of motion.

• The direction of a is that of ∆v.

In both cases shown above, acceleration is positive (i.e. right). In case b, velocity is negative.

A skydiver is falling down (y component negative) at a speed of 48 m/s. She opens her chute. 11 seconds later, she has slowed down to 26 m/s. Her acceleration is:

A. 2.0 m/s/sB. -2.0 m/s/sC. 22 m/s/sD. -22 m/s/s

Equation of a line

What is the equation of this velocity vs time graph? What quantity is represented by the slope?

Velocity as a function of time

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sm 12 Slope

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The slope of a linear velocity vs time graph is the acceleration

The slope of a linear velocity vs time graph is the acceleration

v = 6m/s2 (Δt) + 5m/s