Rectilinear Motion

AP Calculus 2015

Describe motion in a straight line

Which car has the greatest velocity? BLUE

Which car has a constant velocity? RED

Which 2 cars have the same average velocity?

RED and GREEN

Rectilinear Motion

ONE OF THE most important applications of calculus is to motion in a straight line, which is called rectilinear motion.

Consider a point P moving in a straight line. Let s be the position measured from a fixed point O

to any position of P, and let t be the time elapsed. Then to each value of t there will correspond a

position s. s will be a function of t: s = f(t) is the position function of the particle The graph of s versus t is the position versus time

curve.

Example 1

The following figure shows the position versus time curve for a jackrabbit moving along an s-axis. In words, describe how the position of the rabbit changes with time.

position

time

Example 1 Solution

When t=0 Thumper is at s=-3. He moves in the positive direction until t=4

and he is at s=4. Thumper then changes to the negative

direction until t=6.2 and he is at s=-1. Thumper then stops at s=-1 from t=6.2 to

t=10.

s

-3 1-1 4

0t 4t

6.2t

Position, Velocity, & Acceleration

is the position function of a particle moving on a coordinate plane

is the instantaneous velocity of the particle at time t

is the instantaneous acceleration of the particle at time t

s t

'v t s t

' "a t v t s t

time

position

acc posvel pos &increasing

acc zerovel pos &constant

acc negvel pos &decreasing

velocityzero

acc negvel neg &decreasing acc zero

vel neg &constant

acc posvel neg &increasing

acc zero,velocity zero

It is important to understand the relationship between a position graph, velocity and acceleration:

Speeding Up/Slowing Down

A particle in rectilinear motion is SPEEDING UP when its velocity and acceleration have the SAME SIGN

A particle in rectilinear motion is SLOWING DOWN when its velocity and acceleration have OPPOSITE SIGNS.

Observations: Position vs Time

Complete the following Chart

Position vs Time Velocity/Acceleration Particle Behavior

Displacement vs Distance Traveled

The displacement describes the change in position of the particle. Where did the particle end up in relationship to where it started?

Distance traveled is how much total distance the particle traveled. This includes both positive and negative distance. Think about it in terms of when is the particle moving in the negative (down or to the left) direction and when in the positive (up or to the right) direction.

Example 2-Thumper Encore

For the jackrabbit moving along an s-axis, when is Thumper slowing down and when is he speeding up?

What was Thumper’s displacement?

How much distance did he travel?

Example 3

A particle moves according to the function where t is measured in

seconds and s(t) is measured in feet. Find the velocity function What is the velocity at 3 seconds? When is the particle at rest? When is the particle moving forward? Find the total distance traveled during the first 8

seconds. Find the displacement during at 8 seconds. Find the acceleration at 3 seconds. When is the particle speeding up? Slowing down?

3 212 36 , 0s t t t t t

23 24 36v t t t 3 9v

2, 6t t 0,2 & 6,on

96 .ft

3 6a

32 .ft

Example con’t

Interval Velocity Sign Acceleration Sign

Particle Behavior

(0, 2) V(t) A(t)

(2, 4) V(t) A(t)

(4, 6) V(t) A(t)

(6, ) V(t) A(t)

When is the particle speeding up? Slowing down?

Free-Fall Motion

Free-fall motion is the motion that occurs when an object near the Earth is imparted some initial vertical velocity and thereafter moves on a vertical line.

The only force acting on the object is the Earth’s gravity and the gravitational force is constant.

GravitationalConstants:

2

ft32

secg

2

m9.8

secg

2

cm980

secg

Height of an Object

Given the positive direction of the s-axis is up, and the origin is at the Earth’s surface then at any time t the height of an object is given by

where s0 is the initial height, v0 is the initial velocity and g is the acceleration due to gravity. Depending on the units of measure

20 0

1

2s t s v t gt

2 29.8 or 32 ftmg gs s

Example 3

Nolan Ryan, one of the fastest baseball pitchers of all time, was capable of throwing a baseball 150 feet per second. During his career, he had the opportunity to pitch in the Houston Astrodome, an indoor stadium with a ceiling of 208 feet. Could Nolan Ryan have hit the ceiling of the Astrodome if he were capable of giving a baseball an upward velocity of 100 feet per second from a height of 7 feet? 216 100 7s t t t

Example 4

Stan Dup is frustrated after a Calculus test, and, after leaving math class, tosses his textbook vertically into the air (in hopes it never returns to the ground). He releases the text from a point 2 meters above the ground with an initial velocity of 10 m/s. How far above the ground is the textbook after 2

seconds? How fast it is going at the end of 2 seconds? What is the highest point above the ground that the

textbook will reach? What is the average velocity of the textbook between

t =1 and t = 2?

HWQ 10/30/14

A particle is moving along a horizontal line with position function .

Determine the intervals on which the particle is speeding up or slowing down.

Speeding up on [2,3] and [4,infinity], Slowing down on [0,2], and [3,4]

Determine the distance traveled by the particle on [0,5].

Particle travels 28 units

Determine the displacement of the particle at t=5. Displacement is 20 units

3 29 24 4s t x x x

Homework

(Day 1) Motion Module WS pgs. 7-8

(Day 2) MMM pg. 79-81

(Day 3) Position, Velocity, Acceleration PKT