Name _______________________________
Particle Motion Recap & Review
Ex. 1: A particle moves on a number line so that, t seconds after it starts to move, is
position in feet per second is x(t )  2t 3  6t 2  4 .
a. Find an equation for the velocity at any time, t.
b. Find when the particle moves through the zero position for the first time and what
its velocity is at that time.
c. Find when and where it stops.
d. Over what intervals is the particle moving left? moving right?
e. During what interval of time is the particle’s velocity increasing?
f. During what interval of time is the particle’s speed increasing?
g. Compare and contrast the total distance and the total displacement traveled
during the first 3 seconds.
h. Find the average velocity during the first 4 seconds.
i. Find the average acceleration during the first 4 seconds.
1. A particle moves along the x-axis so that at time t  0 its position is given by
x(t )  t 4  2t 3  t 2  2 . At the instant when the acceleration becomes zero, what is
the velocity of the particle?
2. A particle moves along the x-axis so that it’s position for 0  t  5 is given by
x(t )  t 4  15t 3  44t 2  36t  2 . When is the speed the greatest?
3. A particle moves along the x-axis is such a way that its position at time t  0 is
given by the formula x(t )  3t 5  25t 3  60t . For what values of t is the particle
moving to the left?
4. Find when the speed is increasing if x(t )  2t 3  21t 2  60t  8 . Justify.
5. Find when the speed is increasing if x(t )  2t  t 2 . Justify. (Calculator)
6. A particle moves along the x-axis is such a way that its position at time t  0 is
1
14
given by the formula x(t )  t 4  t 3  11t 2  10t  3 , where x is measured in feet and
2
3
t is measured in seconds.
a. Find it’s position after 2 seconds.
b. Find the time during which it has increasing velocity.
c. Find the times during which it is moving to the right.
d. Find the times and positions when the particle is at rest.
e. Find the times and positions when the particle changes direction.
f. Find the total distance traveled in the first 5 seconds.
g. Find the time when the instantaneous velocity equals the average velocity
for the first 5 seconds.
7. Suppose that s(t )  t 2  t  4 is the position function for a particle. Find the value of
t in [0,3] where the instantaneous velocity is equal to the average velocity.
8. The motion of a swimmer playing in the water swimming straight out (or in) from a
point on the shore is described by the formula s (t )  5  t  4   t  1  70 for 0  t  4 .
2
When is the velocity of the swimmer negative?
9. At t  2 , which of the following statements is/are true for the motion described by
s(t )  t 3  t  12 (choose all that apply):
a. The particle is moving toward the zero position.
b. The speed of the particle is increasing.
c. The velocity of the particle is decreasing.
1
1
10. A particle moves in a horizontal line according to the law x(t )  t 
.
2 1 t2
a. Find the initial velocity.
b. Does the particle stop? If so, when?
c. Find the average velocity between time t  1 sec and t  2 sec.
d. Find the velocity at t  2 sec.
11. Let s(t )  t 3 . For what value(s) of t in the interval [0,3] does the average velocity
equal the instantaneous velocity?
12. A particle moves so that its position is given by x(t )   t  2   t  4  for 0  t  4 .
2
a.
b.
c.
d.
2
When and where does the particle stop?
Does is change direction at the times it stops?
Is there a maximum velocity? If so, when does it occur?
Is there a minimum acceleration? If so, what is it?
13. A particle moves along a horizontal line so that its velocity is given by
v(t )  2  t  2  t  6  for 0  t  8 .
a. Find the velocity at 3 seconds.
b. Find the minimum acceleration.
c. Find the average acceleration of the particle over the interval [0,5] sec.
14. A particle moves so that its position is given by x(t )  t 4  8t 3  18t 2  5t  6 . Find the
times when the velocity is increasing. Justify.
15. What is the maximum velocity attained on the interval [0,3] sec by a particle whose
position is given by s(t )  t 3  6t 2  9t  1?
16. A particle, P, moves along a horizontal line. The graph shows its position as a
function of time.
a. When is P moving to the left?
b. When is P standing still?
c. When is the first time that P reverses
direction?
d. When does P move at its greatest speed?
17. A particle, P moves along a horizontal line. The graph shows its velocity as a
function of time.
a. When is P moving to the left?
b. When is P standing still?
c. When is the first time that P reverses
direction?
d. When does P move at its greatest speed?