1. 3.4 Velocity & Other Rates of Change
Annecy, French Alps
3.4 Velocity & Other Rates of Change

2. Slopes of Displacement Functions
The slope of a function is equal to the rate of change of that function:
Slope = rate of change =
Let f(t) be a function that describes how far an object is from its starting point (displacement)

3. Velocity versus Speed
In Calculus velocity and speed are not the same thing.
Instantaneous Velocity:
Velocity includes direction.
Speed:
Speed does not include direction.

4. Velocity in Graphs
S(t)
M = Average Speed

5. Freefall Constants
On the planet earth, the gravitational force is constant.
In meters
G= 9.8 meters per second per second
In feet:
G= 32 feet per second per second

6. Freefall on Earth
The formulas for freefall are shown below. They are used to calculate the position (s) of any object at any given time (t).
In meters
In feet:
Initial Velocity
Initial Position

7. Freefall and Derivatives
Using the formula for freefall, formulas for velocity and acceleration can be quickly computed.

8. Freefall and Derivatives
A penny is dropped off the top of the Eiffel Tower (972 ft). When it hits the ground, how fast is it traveling?

9. Freefall and Derivatives
A penny is thrown straight up off the top of the Eiffel Tower (972 ft). It initial speed is 64 feet per second. When it hits the ground, how fast is it traveling?

10. Freefall and Derivatives
A penny is thrown straight down off the top of the Eiffel Tower (972 ft). It initial speed is 64 feet per second. When it hits the ground, how fast is it traveling?

11. A person on ground level throws an object upwards such that the height, h, in feet at time t, in seconds, is given by h(t) = -16t2 + 68t.
Find the velocity and acceleration.
Determine the displacement over the interval [1, 4].
Determine the average velocity over the interval [1, 4].
Determine when the object hits the ground.
Find the velocity and acceleration of the object at the time it hits the ground.

12. Particle Motion Problems
Atomic particle are subject to many different forces besides gravity. Therefore, the motion of a particle can take a multitude of forms. To limit the possibilities, we will only deal with particles that travel forward & backward on a line.
Example:
A particle is moving along a line. The position of the particle can be described by the equation:
Describe the motion of the particle.

13. Particle Motion Problems
A particle is moving along a line. The position of the particle can be described by the equation:
Describe the motion of the particle.
Here is a graph of s(t):

14. Particle Motion Problems
A particle is moving along a line. The position of the particle can be described by the equation:

15. Particle Motion Problems
A particle is moving along a line. The position of the particle can be described by the equation:

16. Particle Motion Problems
A graph of a particle velocity is shown.
1.) What does the graph of position look like?
2.) What does the graph of acceleration look like?

17. Particle Motion Problems
A graph of a particle velocity is shown.
1.) What does the graph of position look like?
2.) What does the graph of acceleration look like?

18. Particle Motion Problems
A graph of a particle acceleration is shown.
1.) What does the graph of velocity look like?
2.) What does the graph of position look like?

19. A particle moves along the x-axis and its position
at time t is given by s(t) = t3 – 12t2 + 27t,
where t is measured in seconds and s in feet.
Find the velocity and acceleration.
Determine the velocity and acceleration after 5 seconds.
Determine when the particle is at rest.
Determine when the particle is moving forward.
Find the displacement of the particle during the first 6 seconds.
Determine the velocity of the particle when there is no acceleration.

20. A particle moves along the x-axis and its position
at time t is given by ,
where t is measured in seconds and s in feet.
Find the velocity and acceleration.
Determine the velocity and acceleration after 5 seconds.
Determine when the particle is at rest.
Determine when the particle is moving forward.
Find the displacement of the particle during the first 6 seconds.
Determine the acceleration of the particle when velocity is at a minimum.