1. Velocity, Acceleration, Jerk
Section 3.4a

2. Definition: Instantaneous Rate of Change
The difference quotient:
When we let h approach 0, we saw the rate at which a function
was changing at a particular point x…
Definition: Instantaneous Rate of Change
The (instantaneous) rate of change of f with respect to x at a
is the derivative
provided the limit exists.

3. Example 1: Enlarging Circles
(a) Find the rate of change of the area A of a circle with respect
to its radius r.
(b) Evaluate the rate of change of A at r = 5 and at r = 10.
(c) If r is measured in inches and A is measured in square inches,
what units would be appropriate for dA/dr ?
Instantaneous rate of change of A with respect to r :
Units?
Rate at r = 5:
Square inches (of area)
per inch (of radius)
Rate at r = 10:
The rate of change gets bigger as r gets bigger!!!

4. Motion Along a Line
If an object is moving along an axis, we may know its position s on
that line as a function of time t :
The displacement of the object over the interval t to t + t :
The average velocity of the object over this time interval:
How would we find the object’s velocity at the exact instant t ?

5. Definition: Instantaneous Velocity
The (instantaneous) velocity is the derivative of the position
function s = f(t) with respect to time. At time t the velocity is
Definition: Speed
Speed is the absolute value of velocity.
Speed =

6. Example 2: Reading a Velocity Graph
A particle moves along an axis, and its velocity is shown in the
graph below. When does the particle have maximum speed ?
v (m/sec)
The particle moves forward for the first 10 seconds,
then moves backward for the next 8 seconds, stands
still for 4 seconds, and then moves forward again.
The particle achieves
its maximum speed at
about t = 15, while
moving backward. 5
t (sec)

7. Definition: Acceleration
Acceleration is the derivative of velocity with respect to time.
If a body’s velocity at time t is v(t) = ds/dt, then the body’s
acceleration at time t is
Definition: Jerk
Jerk is the derivative of acceleration with respect to time. If a
body’s position at time t is s(t), the body’s jerk at time t is

8. Free-Fall Constants (Earth)
English units:
(s in feet)
Metric units:
(s in meters)

9. Example 3: Modeling Vertical Motion
A dynamite blast propels a heavy rock straight up with a launch
velocity of 160 ft/sec (about 109 mph). It reaches a height of
after t seconds.
(a) How high does the rock go?
Model the situation: s
In our model, velocity is positive on the
way up, and negative on the way down. s
v = 0
max
Find velocity at any time t:
Height (ft)
ft/sec
The velocity is zero when:
sec
s = 0

10. Example 3: Modeling Vertical Motion
A dynamite blast propels a heavy rock straight up with a launch
velocity of 160 ft/sec (about 109 mph). It reaches a height of
after t seconds.
(a) How high does the rock go?
Model the situation: s
The maximum height of the rock is the
height at t = 5 sec: s
v = 0
max
Height (ft) ft
s = 0

11. Example 3: Modeling Vertical Motion
A dynamite blast propels a heavy rock straight up with a launch
velocity of 160 ft/sec (about 109 mph). It reaches a height of
after t seconds.
(b) What is the velocity and speed of the
rock when it is 256 ft above the ground
on the way up? On the way down? s s
v = 0
max
256
t = ?
Height (ft)
ft/sec
ft/sec
s = 0
At both instants, the speed of the rock is 96 ft/sec

12. Example 3: Modeling Vertical Motion
A dynamite blast propels a heavy rock straight up with a launch
velocity of 160 ft/sec (about 109 mph). It reaches a height of
after t seconds. s
(c) What is the acceleration of the rock at
any time t during its flight? s
v = 0
max
256
t = ?
Height (ft)
The acceleration is always negative!!!
s = 0

13. Example 3: Modeling Vertical Motion
A dynamite blast propels a heavy rock straight up with a launch
velocity of 160 ft/sec (about 109 mph). It reaches a height of
after t seconds. s
(d) When does the rock hit the ground? s
v = 0
max
256
t = ?
Height (ft)
The rock hits the ground
10 seconds after the blast.
Let’s graph the position, velocity, and
acceleration functions together in the
same window: [0,10] by [–160,400].
s = 0

14. Example 4: A Moving Particle
A particle moves along a line so that its position at any time
is given by the function
where s is measured in meters and t is measured in seconds.
(a) Find the displacement during the first 5 seconds.
(b) Find the average velocity during the first 5 seconds.
(c) Find the instantaneous velocity when t = 4.
(d) Find the acceleration of the particle when t = 4.
(e) At what values of t does the particle change direction?
(f) Where is the particle when s is a minimum?

15. Example 4: A Moving Particle
A particle moves along a line so that its position at any time
is given by the function
where s is measured in meters and t is measured in seconds.
(a) Find the displacement during the first 5 seconds.
Displacement =
(b) Find the average velocity during the first 5 seconds.
Average Velocity =
(c) Find the instantaneous velocity when t = 4.

16. Example 4: A Moving Particle
A particle moves along a line so that its position at any time
is given by the function
where s is measured in meters and t is measured in seconds.
(d) Find the acceleration of the particle when t = 4.
Acceleration =
(e) At what values of t does the particle change direction?
(f) Where is the particle when s is a minimum?
Since acceleration is always positive,
the position s is at a minimum when
the particle changes direction, at t = 3/2.