Motion Problems—You must show work on most of these problems (sign tests, etc.) These are
sample AP problems, so please do them in a way that will be useful for you to study from in the
spring. You should not be using calculators for any of these problems (except maybe for
arithmetic—no graphing or calculus features.)
1) The table shows the velocity at time t of an object moving along a line. Estimate the
acceleration in ft/sec2 at t = 6 sec.
t (sec)
velocity
(A) -6
0
18
(B) -1.8
4
16
(C) -1.5
8
10
(D) 1.6
(E) 6
Use the graph shown for questions 2-8. It shows the velocity of an
object moving along a straight line during the time interval 0 ≤ t ≤ 5.
2) The object attains its maximum speed when t =
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5
3) The speed of an object is increasing during the time interval
(A) (0, 1)
(B) (1, 2)
(C) (0, 2)
(D) (2, 3)
(E) (3, 5)
4) The acceleration of an object is positive during the time interval
(A) (0, 1)
(B) (1, 2)
(C) (0, 2)
(D) (2, 3)
(E) (3, 5)
5) How many times on 0 < t < 5 is the object's acceleration undefined?
(A) none
(B) 1
(C) 2
(D) 3
(E) more than 3
6) During 2 < t < 3 the object's acceleration (in ft/sec2) is
(A) -10
(B) -5
(C) 0
(D) 5
(E) 10
(D) 3
(E) 5
7) The object is furthest to the right when t =
(A) 0
(B) 1
(C) 2
8) The object's average acceleration (in ft/sec2) for the interval 0 ≤ t ≤ 3 is
(A) -15
(B) -5
(C) -3
(D) -1
(E) none of these
10
0
9) The table shows the velocity at time t of an object moving along a line. Estimate the
acceleration in ft/sec2 at t = 1 sec.
t (sec)
velocity
(A) 0.8
1.0
12.2
(B) 1.0
1.5
13.0
(C) 1.2
2.0
13.4
(D) 1.4
2.5
13.7
(E) 1.6
In questions 10-13, the position of a particle moving along a straight line is given by
s = t3 – 6t2 + 12t – 8.
10) The position s is increasing for
(A) t < 2
(D) t < 1 or t > 3
(B) all t
(E) t > 2
(C) 1 < t < 3
11) The minimum value of the speed is
(A) 1
(B) 2
(C) 3
(D) 0
(E) none of these
12) The acceleration is positive for
(A) t > 2
(D) 1 < t < 3
(B) all t except t = 2 (C) t < 2
(E) 1 < t < 2
13) The speed of the particle is decreasing for
(A) t > 2
(D) t < 1 or t > 2
(B) t < 3
(E) none of these
(C) all t
In questions 14-16, a particle moves along a horizontal line and its position at time t is
s = t4 – 6t3 + 12t2 + 3.
14) The particle is at rest when t is equal to
(A) 1 or 2
(B) 0
(C) 9/4
(D) 0, 2, 3
15) The velocity is increasing when
(A) t > 1
(D) t < 1 or t > 2
(B) 1 < t < 2
(E) t > 0
(C) t < 2
16) The speed of the particle is increasing for
(A) 0 < t < 1 or t > 2 (B) 1 < t < 2
(D) t < 0 or t > 2
(E) t < 0
(C) t < 2
(E) none of these
17) The displacement from the origin of a particle moving on a line is given by
s = t4 – 4t3. The maximum displacement during the time interval -2 ≤ t ≤ 4 is
(A) 27
(B) 3
(C) 12 3  3
(D) 48
(E) none of these
18) If a particle moves along a line according to the law s = t5 + 5t4, then the number of times it
reverses direction is
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
The graph for questions 19 and 20 show the velocity of an object moving along a straight line
during the time interval 0 ≤ t ≤ 12.
19) For what t does this object attain its maximum
acceleration?
(A) 0<t<4 (B) 4<t<8 (C) 5
(D) 8
(E) 12
20) The object reverses direction at t =
(A) 4 only
(B) 5 only
(C) 8 only
(D) 5, 8
(E) none of these
21) A particle moves along a line in such a way that its position at time t is given by
s = t3 – 6t2 + 9t + 3. Its direction of motion changes when t =
(A) 1 only
(B) 2 only
(C) 3 only
(D) 1, 3
(E) 1, 2, 3
22) A body moves a long a straight line so that its velocity v at time t is given by
v = 4t3 + 3t2 + 5. The distance the body covers from t = 0 to t = 2 equals
(A) 34
(B) 55
(C) 24
(D) 44
(E) none of these
23) A particle moves along a line with velocity v = 3t2 – 6t. The total distance traveled from t =
0 to t = 3 equals
(A) 9
(B) 4
(C) 2
(D) 16
(E) none of these
24) The net change in the position of the particle in question 23 is
(A) 2
(B) 4
(C) 9
(D) 16
(E) none of these
25) The acceleration of a particle moving on a straight line is given by a = cos(t), and when t = 0
the particle is at rest. The distance it covers from t = 0 to t = 2 is
(A) sin(2)
(B) 1 – cos(2)
(D) sin(2) – 1 (E) –cos(2)
(C) cos(2)
26) During the worst 4-hr period of a hurricane the wind velocity, in mph, is given by the
v(t) = 5t – t2 + 100, 0 ≤ t ≤ 4. The average wind velocity during this period is
(A) 10
(B) 100
(C) 102
(D) 104 23
(E)
108 23
27) A car accelerates from 0 to 60 mph in 10 sec, with constant acceleration.
(Note that 60 mph = 88ft/sec.) The acceleration (in ft/sec2) is
(A) 5.3
(B) 6
(C) 8
Answers
1) C
2) D
3) D
4) E
5) D
6) A
7) C
8) B
9) E
10) B
11) D
12) A
13) E (t < 2)
14) B
15) D
16) A
17) D
18) C
19) B
20) B
21) D
22) A
23) E (8)
24) E (0)
25) B
26) D
27) D
(D) 8.8
(E) none of these