1. Rectilinear Motion
AP Calculus 2015

2. Describe motion in a straight line
Which car has the greatest velocity?
BLUE
Which car has a constant velocity?
RED
RED and GREEN
Which 2 cars have the same average velocity?

3. 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.

4. 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

5. 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. -3 1 -1 4

6. 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

7. It is important to understand the relationship between a position graph, velocity and acceleration:
time
position
acc neg
vel pos &
decreasing
acc neg
vel neg &
decreasing
acc zero
vel neg &
constant
acc zero
vel pos &
constant
acc pos
vel neg &
increasing
velocity
zero
acc pos
vel pos &
increasing
acc zero,
velocity zero

8. 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.

9. Observations: Position vs Time
Complete the following Chart
Position vs Time
Velocity/Acceleration
Particle Behavior

10. 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.

11. 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?

12. 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?

13. Example con’t (0, 2) V(t) A(t) (2, 4) (4, 6) (6, ) Interval
When is the particle speeding up? Slowing down?
Interval
Velocity Sign
Acceleration Sign
Particle Behavior
(0, 2)
V(t)
A(t)
(2, 4)
(4, 6)
(6, )

14. 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.
Gravitational
Constants:

15. 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

16. 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?

17. 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?

18. 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

19. Homework (Day 1) Motion Module WS pgs. 7-8 (Day 2) MMM pg. 79-81
(Day 3) Position, Velocity, Acceleration PKT