5128_Ch03_pp098-184

Section 3.4 Velocity and Other Rates of Change 137
18. Launching a Rocket When a model rocket is launched,
the propellant burns for a few seconds, accelerating the rocket
upward. After burnout, the rocket coasts upward for a while
and then begins to fall. A small explosive charge pops out a
parachute shortly after the rocket starts downward. The
parachute slows the rocket to keep it from breaking when it
lands. This graph shows velocity data from the flight.
Velocity (ft/sec)
200
150
100
50
0
–50
–100
0 2 4 6 8 10 12
Time after launch (sec)
Use the graph to answer the following.
190 ft/sec
(a) How fast was the rocket climbing when the engine stopped?
(b) For how many seconds did the engine burn? 2 seconds
(c) When did the rocket reach its highest point? What was its
velocity then? After 8 seconds, and its velocity was 0 ft/sec then
(d) When did the parachute pop out? How fast was the rocket
falling then? After about 11 seconds; it was falling 90 ft/sec then.
(e) How long did the rocket fall before the parachute opened?
About 3 seconds
(f) When was the rocket’s acceleration greatest? When was the
acceleration constant? Just before the engine stopped; from t = 2
to t = 11 while the rocket was in free fall
19. Particle Motion A particle moves along a line so that its
position at any time t 0 is given by the function
st t 2 3t 2,
where s is measured in meters and t is measured in seconds.
(a) Find the displacement during the first 5 seconds. 10 m
(b) Find the average velocity during the first 5 seconds. 2 m/sec
(c) Find the instantaneous velocity when t 4.
5 m/sec
(d) Find the acceleration of the particle when t 4. 2 m/sec 2
(e) At what values of t does the particle change direction?
(f) Where is the particle when s is a minimum? At s 1 4 m
20. Particle Motion A particle moves along a line so that its
position at any time t 0 is given by the function s(t)
t 3 7t 2 14t 8 where s is measured in meters and t is
measured in seconds.
v(t) 3t 2 14t 14
(a) Find the instantaneous velocity at any time t.
a(t) 6t 14
(b) Find the acceleration of the particle at any time t.
(c) When is the particle at rest? t 1.451, 3.215
(d) Describe the motion of the particle. At what values of t does
the particle change directions?
21. Particle Motion A particle moves along a line so that its
position at any time t 0 is given by the function s (t)
(t 2) 2 (t 4) where s is measured in meters and t is measured
in seconds.
v(t) 3t 2 16t 20 (t 2)(3t 10)
(a) Find the instantaneous velocity at any time t.
(b) Find the acceleration of the particle at any time t. a(t) 6t 16
(c) When is the particle at rest? t 2, 10/3
(d) Describe the motion of the particle. At what values of t does
the particle change directions?
22. Particle Motion A particle moves along a line so that its
position at any time t 0 is given by the function s (t)
t 3 6t 2 8t 2 where s is measured in meters and t is
measured in seconds.
v(t) 3t 2 12t 8
(a) Find the instantaneous velocity at any time t.
(b) Find the acceleration of the particle at any time t. a(t) 6t 12
(c) When is the particle at rest? t 0.845, 3.155
(d) Describe the motion of the particle. At what values of t does
the particle change directions?
23. Particle Motion The position of a body at time t sec is
s t 3 6t 2 9t m. Find the body’s acceleration each
time the velocity is zero. At t 1; 6 m/sec 2 At t 3: 6 m/sec 2
24. Finding Speed A body’s velocity at time t sec is
v 2t 3 9t 2 12t 5 msec. Find the body’s speed
each time the acceleration is zero. At t 1: 0 m/sec; at t 2:1 m/sec
25. Draining a Tank It takes 12 hours to drain a storage tank by
opening the valve at the bottom. The depth y of fluid in the tank
t hours after the valve is opened is given by the formula
2
y 6( 1 t
12
) m.
(a) Find the rate dydt mh at which the water level is
changing at time t. d y t
1
dt
1 2
(b) When is the fluid level in the tank falling fastest? slowest?
What are the values of dydt at these times?
At t 3 2 sec
(c) Graph y and dydt together and discuss the behavior of y
in relation to the signs and values of dydt.
20. (d) The particle starts at the point s 8 and moves left until it stops at
s 0.631 at t 1.451, then it moves right to the point s 2.213 at t
3.215,) where it stops again, and finally continues left from there on.
21. (d) The particle starts at the point s 16 when t 0 and moves right
until it stops at s 0 when t 2, then it moves left to the point s 1.185
when t 10/3 where it stops again, and finally continues right from there on.
22. (d) The particle starts at the point s 2 when t 0 and moves right until
it stops at s 5.079 at t 0.845, then it moves left to the point s 1.079 at
t 3.155 where it stops again, and finally continues right from there on.
25. (b) Fastest: at t 0 slowest: at t 12; at t 0: d y
1;
dt
at t = 12: d y
0
dt