BC CALCULUS
PARTICLE MOTION
Work these on notebook paper. A calculator may be used on all problems.
1. A particle moves along a horizontal line so that its position at any time is given by
s  t   t 3  12t 2  36t , t  0, where s is measured in meters and t in seconds. Do not
use your calculator except on part (f).
(a) Find the instantaneous velocity at time t and at t = 3 seconds.
(b) When is the particle at rest? Moving to the right? Moving to the left? Justify your answers.
(c) Find the displacement of the particle after the first 8 seconds.
(d) Find the total distance traveled by the particle during the first 8 seconds.
(e) Find the acceleration of the particle at time t and at t = 3 seconds.
(f) Graph the position, velocity, and acceleration functions for 0  t  8 .
(g) When is the particle speeding up? Slowing down? Justify your answers.
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2. A particle moves along the x-axis with its position at time t given by
t
x t  
, t0
1 t2
where t is measured in seconds and x in meters. Use your calculator.
(a) Find the velocity at time t and at t = 2 seconds.
(b) When is the particle at rest? When is moving to the right? To the left? Justify your answers.
(c) Find the displacement of the particle during the time interval 0  t  4 .
(d) Find the total distance traveled by the particle during the time interval 0  t  4 .
(e) Find the acceleration of the particle at time t and at t = 3 seconds.
(f) When is the particle speeding up? Slowing down? Justify your answers.
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3. The maximum acceleration attained on the interval 0  t  3 by the particle
whose velocity is given by v  t   t 3  3t 2  12t  4 is
(A) 9
(B) 12
(C) 14
(D) 21
(E) 40
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4. 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
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5. The figure on the right shows the position s of a
particle moving along a horizontal line.
(a) When is the particle moving to the left? moving
to the right? standing still? Justify your answer.
(b) For each of v 1.5 , v  2.5 , v  4  , and v  5 , find
the value or explain why it does not exist.
(c) Graph the particle’s velocity.
(d) Graph the particle’s speed.
TURN->>>
Answers
1. (a) 3t 2  24t  36,  9m / sec
(b) At rest at t = 2 because v  t  = 0 there. Moving right for [0, 2) and  6,   because v  t  > 0.
Moving left for (2, 6) because v  t  < 0.
(c) 32 meters
(d) 96 meters
(e) 6t  24,  6m / sec2
(f) Graph
(g) Speeding up on (2, 4) and  6,   because vel. and acc. have the same sign. Slowing down
on (0, 2) and (4, 6) because vel. and acc. have opposite signs.
2. (a)
1 t2
1  t 
2 2
, 
3
m / sec
25
(b) At rest when t = 1 because v  t  = 0 there. Moving right for [0, 1) because v  t  > 0.
Moving left for 1,   because v  t  < 0.
(c) 0.235 m
(d) 0.765 m
2t t 2  3
(e)
, .036 m / sec 2
2 3
1 t




(f) Slowing down on (0, 1) and




3,  because vel. and acc. have opposite signs. Speeding up
on 1, 3 because vel. and acc. have the same sign
3. D
4. C
5. (a) Moving left on (2, 3) and (5, 6) because v  t  < 0. Moving right on (0, 1) because v  t  > 0.
Standing still on (1, 2) and (3, 5) because v  t  = 0 there.
(b) 0, - 4, 0, dne because graph of s has a sharp turn there
(c) and (d) Graphs