Unit 4.2/4.8/5.1: Derivatives, Anti-Derivatives
and the Physics of Motion
Velocity and Acceleration
Suppose that 𝑠(𝑡) provides the position function of an object as a function of time.
The average velocity is the displacement divided by time:
Average Velocity =
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑛 𝑜𝑏𝑗𝑒𝑐𝑡
𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
=
𝑠(𝑡2 )−𝑠(𝑡1 )
𝑡2 −𝑡1
∆𝑠
= ∆𝑡
The instantaneous velocity (or just ‘velocity’ 𝑣(𝑡)) is the derivative of the position function:
Velocity = 𝑣(𝑡) = lim
∆𝑠
∆𝑡→0 ∆𝑡
= 𝑠 ′ (𝑡)
The acceleration 𝑎(𝑡) is the derivative of the velocity function:
Acceleration = 𝑎(𝑡) = lim
∆𝑣
∆𝑡→0 ∆𝑡
= 𝑣 ′ (𝑡) = 𝑠 ′′ (𝑡)
Section 3.1, #27
27. Object dropped from a tower An object is dropped from the top of a 100-m-high tower. Its height above
ground after t sec is 100 − 4.9𝑡 2 m. How fast is it falling 2 sec after it is dropped?
Section 3.4, #1, 3
Exercises 1—3 give the positions 𝑠 = 𝑓(𝑡)of a body moving on a coordinate line, with s in meters and t in
seconds.
1. 𝑠(𝑡) = 𝑡 2 − 3𝑡 + 2,
0≤𝑡≤2
a. Find the body’s displacement and average velocity for the given time interval.
b. Find the body’s speed and acceleration at the endpoints of the interval.
c. When, if ever, during the interval does the body change direction?
2. 𝑠(𝑡) = 6𝑡 − 𝑡 2 ,
0≤𝑡≤6
a. Find the body’s displacement and average velocity for the given time interval.
b. Find the body’s speed and acceleration at the endpoints of the interval.
c. When, if ever, during the interval does the body change direction?
3. 𝑠(𝑡) = −𝑡 3 + 3𝑡 2 − 3𝑡,
0≤𝑡≤3
a. Find the body’s displacement and average velocity for the given time interval.
b. Find the body’s speed and acceleration at the endpoints of the interval.
c. When, if ever, during the interval does the body change direction?
Section 3.4, #9
9. Free fall on Mars and Jupiter The equations for free fall at the surfaces of Mars and Jupiter (s in meters, t
in seconds) are 𝑠 = 1.86𝑡 2 on Mars and 𝑠 = 11.44𝑡 2 on Jupiter. How long does it take a rock falling from rest
to reach a velocity of 27.8 m/sec (about 100 km/h) on each planet?
Section 3.4, #15
𝑑𝑠
15. The accompanying figure shows the velocity 𝑣 = 𝑑𝑡 = 𝑓(𝑡) (m/sec) of a body moving along a coordinate
line.
a. When does the body reverse direction?
b. When (approximately) is the body moving at a constant speed?
c. Graph the body’s speed for 0 ≤ 𝑡 ≤ 10.
d. Graph the acceleration, where defined.
HW 3.6, #99
99. Particle Motion The position of a particle moving along a coordinate line is 𝑠 = √1 + 4𝑡, with s in meters and t in
seconds. Find the particle’s velocity and acceleration at t = 6 sec.
Antiderivatives: We can recover the displacement from the velocity using the antiderivative.
HW 4.2, #43
Exercises 43 -- 44 give the velocity 𝑣 = 𝑑𝑠/𝑑𝑡 and the initial position of a body moving along a coordinate line. Find the
body’s position at time t.
43. 𝑣 = 9.8𝑡 + 5, 𝑠(0) = 10
44. 𝑣 = 32𝑡 − 2, 𝑠(0.5) = 4
HW 4.2 #49
Exercises 49 -- 50 give the acceleration 𝑎 = 𝑑2 𝑠/𝑑𝑡 2 , initial velocity, and the initial position of a body moving along a
coordinate line. Find the body’s position at time t.
49. 𝑎 = −4 sin 𝑡 , 𝑣(0) = 2,
9
3𝑡
50. 𝑎 = 𝜋2 cos 𝜋 , 𝑣(0) = 0,
𝑠(0) = −3
𝑠(0) = −1
HW 4.8 #123
𝑑2 𝑠
123. Motion along a coordinate line A particle on a coordinate line with acceleration 𝑎 = 𝑑𝑡 2 = 15√𝑡 − (3/√𝑡),
subject to the conditions that
𝑑𝑠
𝑑𝑡
= 4 and s = 0 when t = 1. Find
a. the velocity 𝑣 = 𝑑𝑠/𝑑𝑡 in terms of t
b. the position s in terms of t
Area under the curve: We can also recover the displacement (distance traveled) by
calculating the area under the velocity curve.
HW5pt1 #9
9. Distance traveled The accompanying table shows the velocity of a model train engine moving along a track
for 10 sec.
Time (sec)
0
1
2
3
4
5
6
7
8
9
10
Velocity (in./sec)
0
12
22
10
5
13
11
6
2
6
0
Estimate the distance traveled by the engine using 10 subintervals of length 1 with
(a) left-endpoint values.
(b) right-endpoint values.
interval
0-1
1-2
2-3
3-4
4-5
5-6
6-7
7-8
8-9
9-10
Distance Traveled
(using left-endpoint)
Distance Traveled
(using right-endpoint)
HW 5pt1 #11
interval
0-10
10-20
20-30
30-40
40-50
50-60
60-70
70-80
80-90
90-100
100-110
110-120
Distance Traveled
(using leftendpoint)
Distance Traveled
(using rightendpoint)