Review of PVA Problems
= class problem
= homework problem
1969 #19
A point moves on the x-axis in such a way that its velocity at time t (t > 0) is given by v 
ln t
. At
t
what value of t does v attain its maximum?
3
1
a) 1
b) e 2
c) e
d) e 2
e) There is no maximum value for v
1969 #35
At t = 0 a particle starts at rest and moves along a line in such a way that at time t its acceleration is
24t2 feet per second per second. Through how many feet does the particle move during the first 2
seconds?
a) 32
b) 48
c) 64
d) 96
e) 192
1973 #8
A particle moves in a straight line with velocity v(t) = t2. How far does the particle move between
times t = 1 and t = 2?
a)
1
3
b)
7
3
c) 3
d) 7
e) 8
1973 #13
The acceleration  of a body moving in a straight line is given in terms of time t by   8  6t . If
the velocity of the body is 25 at t = 1 and if s(t) is the distance of the body from the origin at time t,
what is s(4) – s(2)?
a) 20
b) 24
c) 28
d) 32
e) 42
1973 #28
A point moves in a straight line so that its distance at time t from a fixed point of the line is 8t – 3t2.
What is the total distance covered by the point between t = 1 and t = 2?
a) 1
b)
4
3
c)
5
3
d) 2
e) 5
1985 #11
The position of a particle moving along a straight line at any time t is given by s(t) = t2 + 4t + 4.
What is the acceleration of the particle when t = 4?
a) 0
b) 2
c) 4
d) 8
e) 12
1985 #14
1
3
The velocity of a particle moving on a line at time t is v  3t 2  5t 2 meters per second. How many
meters did the particle travel from t = 0 to t = 4?
a) 32
b) 40
c) 64
d) 80
e) 184
1985 #28
If the position of a particle on the x-axis at time t is -5t2, then the average velocity of the particle for
0 ≤ t ≤ 3 is
a) -45
b) -30
c) -15
d) -10
e) -5
1988 #3
A particle with velocity at any time t given by v(t) = et moves in a straight line. How far does the
particle move from t = 0 to t = 2?
e3
2
2
a) e – 1
b) e – 1
c) 2e
d) e
e)
3
1988 #16
A particle moves along the x-axis so that at any time t ≥ 0 its position is given by
x(t) = t3 – 3t2 – 9t + 1. For what values of t is the particle at rest?
a) no values
b) 1 only
c) 3 only
d) 5 only
e) 1 and 3
1993 #11
The acceleration of a particle moving along the x-axis at any time t is given by a(t) = 6t – 2. If the
velocity is 25 when t = 3 and the position is 10 when t = 1, then the position x(t) =
a) 9t2 + 1
b) 3t2 – 2t + 4
c) t3 – t2 + 4t + 6
d) t3 – t2 + 9t – 20
e) 36t3 – 4t2 – 77t + 55
1993 #26 (calculator question)
A particle moves along a line so that at time t, where 0 ≤ t ≤ π, its position is given by
t2
s(t )  4 cos t   10 . What is the velocity of the particle when its acceleration is zero?
2
a) -5.19
b) 0.74
c) 1.32
d) 2.55
e) 8.13
1997 #8, 9
y
3
2
1
O
-1
1
2
3 4
5
6
7
8
t
A bug begins to crawl up a vertical wire at time t = 0. The velocity v of the bug at time t, 0  t  8,
is given by the function whose graph is shown above.
8. At what value of t does the bug change direction?
a) 2
b) 4
c) 6
d) 7
e) 8
9. What is the total distance the bug traveled from t = 0 to t = 8?
a) 14
b) 13
c) 11
d) 8
e) 6
1997 #87 (calculator question)
At time t ≥ 0, the acceleration of a particle moving on the x-axis is a(t) = t + sint. At t = 0, the
velocity of the particle is -2. For what value t will the velocity of the particle be zero?
a) 1.02
b) 1.48
c) 1.85
d) 2.81
e) 3.14
1998 #14
A particle moves along the x-axis so that its position at time t is given by x(t) = t2 – 6t + 5. For what
value of t is the velocity of the particle zero?
a) 1
b) 2
c) 3
d) 4
e) 5
1998 #24
The maximum acceleration attained on the interval 0 ≤ t ≤ 3 by the particle whose velocity is given
by v(t) = t3 – 3t2 + 12t + 4 is
a) 9
b) 12
c) 14
d) 21
e) 40
2003 #25
A particle moves along the x-axis so that at time t ≥ 0 its position is given by
x(t) = 2t3 – 21t2 + 72t – 53. At what time t is the particle at rest?
a) t = 1 only
b) t = 3 only
c) t =
7
2
only
d) t = 3 and t =
7
2
e) t = 3 and t = 4
2003 #76 (calculator question)
A particle moves along the x-axis so that at any time t ≥ 0, its velocity is given by
v(t) = 3 + 4.1cos(0.9t). What is the acceleration of the particle at time t = 4?
a) -2.016
b) -0.677
c) 1.633
d) 1.814
e) 2.978
2003 #83 (calculator question)
The velocity, in ft/sec, of a particle moving along the x-axis is given by the function v(t) = et + tet.
What is the average velocity of the particle from time t = 0 to time t = 3?
a) 20.086 ft/sec
b) 26.447 ft/sec
c) 32.809 ft/sec
d) 40.671 ft/sec
e) 79.342 ft/sec
2003 #91 (calculator question)
A particle moves along the x-axis so that at any time t > 0, its acceleration is given by
a(t) = ln(1 + 2t ). If the velocity of the particle is 2 at time t = 1, then the velocity of the particle at
time t = 2 is
a) 0.462
b) 1.609
c) 2.555
d) 2.886
e) 3.346
FREE RESPONSE
1989 #3
A particle moves along the x-axis in such a way that its acceleration at time t for t ≥ 0 is given by
a(t) = 4cos(2t). At time t = 0, the velocity of the particle is v(0) = 1 and its position is x(0) = 0.
a) Write an equation for the velocity v(t) of the particle.
b) Write an equation for the position x(t) of the particle.
c) For what values of t, 0 ≤ t ≤ π, is the particle at rest?
1992 #2
A particle moves along the x-axis so that its velocity at time t, 0 ≤ t ≤ 5, is given by
v(t) = 3(t – 1)(t – 3). At time t = 2, the position of the particle is x(2) = 0.
a) Find the minimum acceleration of the particle.
b) Find the total distance traveled by the particle.
c) Find the average velocity of the particle over the interval 0 ≤ t ≤ 5.
1993 #2 (calculator question)
A particle moves on the x-axis so that its position at any time t ≥ 0 is given by x(t) = 2te-t.
a) Find the acceleration of the particle at t = 0.
b) Find the velocity of the particle when its acceleration is 0.
c) Find the total distance traveled by the particle from t = 0 to t = 5.
1994 #4
A particle moves along the x-axis so that at any time t > 0 its velocity is given by v(t) = t lnt – t. At
any time t = 1, the position of the particle is x(1) = 6.
a) Write an expression for the acceleration of the particle.
b) For what values of t is the particle moving to the right?
c) What is the minimum velocity of the particle? Show the analysis that leads to your
conclusion.
d) Write an expression of the position x(t) of the particle.
1995 #2 (calculator question)
A particle moves along the y-axis so that its velocity at any time t ≥ 0 is given by v(t) = tcost. At
any time t = 0, the position of the particle is y = 3.
a) For what values of t, 0 ≤ t ≤ 5, is the particle moving upward?
b) Write an expression for the acceleration of the particle in terms of t.
c) Write an expression for the position y(t) of the particle.
d) For t > 0, find the position of the particle the first time the velocity of the particle is zero.
1997 #1 (calculator question)
A particle moves along the x-axis so that its velocity at any time t ≥ 0 is given by v(t) = 3t2 – 2t – 1.
The position x(t) is 5 for t = 2.
a) Write a polynomial expression for the position of the particle at any time t ≥ 0.
b) For what values of t, 0 ≤ t ≤ 3, is the particle's instantaneous velocity the same as its average
velocity on the closed interval [0, 3]?
c) Find the total distance traveled by the particle from time t = 0 until time t = 3.
1997 #6 (calculator question)
Let v(t) be the velocity in feet per second, of a skydiver at time t seconds, t ≥ 0. After her parachute
dv
 2v  32 , with initial condition
opens, her velocity satisfies the differential equation
dt
v(0) = -50.
a) Use separation of variables to find an expression for v in terms of t, where t is measured in
seconds.
b) Terminal velocity is defined as lim v(t ) . Find the terminal velocity of the skydiver to the
t 
nearest foot per second.
c) It is safe to land when her speed is 20 feet per second. At what time t does she reach this
speed?
Velocity of Runner A
(meters per second)
2000 #2 (calculator question)
(3, 10)
(10, 10)
Time (seconds)
Two runners, A and B, run on a straight racetrack for 0 ≤ t ≤ 10 seconds. The graph above, which
consists of two line segments, shows the velocity, in meters per second, of Runner A. The velocity,
24t
in meters per second, of Runner B is given by the function v defined by v(t ) 
.
2t  3
a) Find the velocity of Runner A and the velocity of Runner B at time t = 2 seconds. Indicate
units of measure.
b) Find the acceleration of Runner A and the acceleration of Runner B at time t = 2 seconds.
Indicate units of measure.
c) Find the total distance run by Runner A and the total distance run by Runner B over the time
interval 0 ≤ t ≤ 10 seconds. Indicate units of measure.
2001 #3 (calculator question)
a(t)
(ft/sec2)
(2, 15)
15
(18, 15)
t (seconds)
2
-15
4
6
8
(10, -15)
10
12
14
16
18
(14, -15)
A car is traveling on a straight road with velocity 55 ft/sec at time t = 0. For 0 ≤ t ≤ 18 seconds, the
car's acceleration a(t), in ft/sec2, is the piecewise linear function defined by the graph above.
a) Is the velocity of the car increasing at t = 2 seconds? Why or why not?
b) At what time in the interval 0 ≤ t ≤ 18, other than t = 0, is the velocity of the car 55 ft/sec?
Why?
c) On the time interval 0 ≤ t ≤ 18, what is the car's absolute maximum velocity, in ft/sec, and at
what time does it occur? Justify your answer.
d) At what times in the interval 0 ≤ t ≤ 18, if any, is the car's velocity equal to zero? Justify
your answer.
2002 #3 (calculator question)
An object moves along the x-axis with initial position x(0) = 2. The velocity of the object at time
 
t ≥ 0 is given by v(t )  sin  t  .
3 
a) What is the acceleration of the object at time t = 4?
b) Consider the following two statements.
Statement I: For 3 < t < 4.5, the velocity of the object is decreasing
Statement II: For 3 < t < 4.5, the speed of the object is increasing
Are either or both of these statements correct? For each statement, provide a reason why it
is correct or not correct.
c) What is the total distance traveled by the object over the time interval 0 ≤ t ≤ 4 ?
d) What is the position of the object at time t = 4?
2003 #2 (calculator question)
𝑡2
A particle moves along the x-axis so that its velocity at time t is given by 𝑣(𝑡) = −(𝑡 + 1) sin ( 2 ).
At time t = 0, the particle is at position x = 1.
a) Find the acceleration of the particle at time t = 2. Is the speed of the particle increasing at t
= 2? Why or why not?
b) Find all times t in the open interval 0 < t < 3 when the particle changes direction. Justify
your answer.
c) Find the total distance traveled by the particle from time t = 0 until time t = 3.
d) During the time interval 0 ≤ t ≤ 3, what is the greatest distance between the particle and the
origin? Show the work that leads to your answer.
2004 #3 (calculator question)
A particle moves along the y-axis so that its velocity v at time t ≥ 0 is given by v(t )  1  tan 1 e t .
At time t = 0, the particle is at y = -1. (Note: tan-1x = arctanx)
 
a) Find the acceleration of the particle at time t = 2.
b) Is the speed of the particle increasing or decreasing at time t = 2? Give a reason for your
answer.
c) Find the time t ≥ 0 at which the particle reaches its highest point. Justify your answer.
d) Find the position of the particle at time t = 2. Is the particle moving toward the origin or
away from the origin at time t = 2? Justify your answer.
v(t)
Velocity
(meters per second)
2005 #5
20
(4, 20)
(16, 20)
15
10
5
t
4
8 12 16 20
Time (seconds)
24
A car is traveling on a straight road. For 0 ≤ t ≤ 24 seconds, the car's velocity v(t), in meters per
second, is modeled by the piecewise-linear function defined by the graph above.
24
24
a) Find ∫0 𝑣(𝑡)𝑑𝑡. Using correct units, explain the meaning of ∫0 𝑣(𝑡)𝑑𝑡.
b) For each of v'(4) and v'(20), find the value or explain why it does not exist. Indicate units of
measure.
c) Let a(t) be the car's acceleration at time t, in meters per second per second. For 0 < t < 24,
write a piecewise-defined function for a(t).
d) Find the average rate of change of v over the interval 8 ≤ t ≤ 20. Does the Mean Value
Theorem guarantee a value of c, for 8 < c < 20, such that v'(c) is equal to this average rate of
change? Why or why not?
2006 #4
t
0
10
20
30
40
50
60
70
80
(seconds)
v(t)
5
14
22
29
35
40
44
47
49
(feet per second)
Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at
time t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0
≤ t ≤ 80 seconds, as shown in the table above.
a) Find the average acceleration of rocket A over the time interval 0 ≤ t ≤ 80 seconds. Indicate
units of measure.
70
b) Using correct units, explain the meaning of ∫10 𝑣(𝑡)𝑑𝑡 in terms of the rocket's flight. Use a
70
midpoint Riemann sum with 3 subintervals of equal length to approximate ∫10 𝑣(𝑡)𝑑𝑡.
3
c) Rocket B is launched upward with an acceleration of a(t ) 
feet per second per second.
t 1
At time t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet
per second. Which of the two rockets is traveling faster at time t = 80 seconds? Explain your
answer.