Circular Motion and Other Applications of Newton's Laws

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time, from the instant it is released until 99% of terminal
speed is reached. (Hint: Try �t � 0.005 s.)
41. A hailstone of mass 4.80 � 10 �4 kg falls through the air
and experiences a net force given by
F ��mg � Cv 2
where C � 2.50 � 10�5 kg/m. (a) Calculate the terminal
speed of the hailstone. (b) Use Euler’s method of
numerical analysis to find the speed and position of the
hailstone at 0.2-s intervals, taking the initial speed to be
zero. Continue the calculation until the hailstone
reaches 99% of terminal speed.
42. A 0.142-kg baseball has a terminal speed of 42.5 m/s
(95 mi/h). (a) If a baseball experiences a drag force of
magnitude R � Cv 2 , what is the value of the constant C?
(b) What is the magnitude of the drag force when the
speed of the baseball is 36.0 m/s? (c) Use a computer
to determine the motion of a baseball thrown vertically
upward at an initial speed of 36.0 m/s. What maximum
height does the ball reach? How long is it in
the air? What is its speed just before it hits the ground?
43. A 50.0-kg parachutist jumps from an airplane and falls
with a drag force proportional to the square of the
speed R � Cv 2 . Take C � 0.200 kg/m with the parachute
closed and C � 20.0 kg/m with the chute open.
(a) Determine the terminal speed of the parachutist in
both configurations, before and after the chute is
opened. (b) Set up a numerical analysis of the motion
and compute the speed and position as functions of
time, assuming the jumper begins the descent at
1 000 m above the ground and is in free fall for 10.0 s
before opening the parachute. (Hint: When the parachute
opens, a sudden large acceleration takes place; a
smaller time step may be necessary in this region.)
44. Consider a 10.0-kg projectile launched with an initial
speed of 100 m/s, at an angle of 35.0° elevation. The resistive
force is R ��bv, where b � 10.0 kg/s. (a) Use a
numerical method to determine the horizontal and vertical
positions of the projectile as functions of time.
(b) What is the range of this projectile? (c) Determine
the elevation angle that gives the maximum range for
the projectile. (Hint: Adjust the elevation angle by trial
and error to find the greatest range.)
45. A professional golfer hits a golf ball of mass 46.0 g with
her 5-iron, and the ball first strikes the ground 155 m
(170 yards) away. The ball experiences a drag force of
magnitude R � Cv and has a terminal speed of
44.0 m/s. (a) Calculate the drag constant C for the golf
ball. (b) Use a numerical method to analyze the trajectory
of this shot. If the initial velocity of the ball makes
an angle of 31.0° (the loft angle) with the horizontal,
what initial speed must the ball have to reach the 155-m
distance? (c) If the same golfer hits the ball with her 9iron
(47.0° loft) and it first strikes the ground 119 m
away, what is the initial speed of the ball? Discuss the
differences in trajectories between the two shots.
2
Problems 177
ADDITIONAL PROBLEMS
46. An 1 800-kg car passes over a bump in a road that follows
the arc of a circle of radius 42.0 m as in Figure
P6.46. (a) What force does the road exert on the car as
the car passes the highest point of the bump if the car
travels at 16.0 m/s? (b) What is the maximum speed the
car can have as it passes this highest point before losing
contact with the road?
47. A car of mass m passes over a bump in a road that follows
the arc of a circle of radius R as in Figure P6.46.
(a) What force does the road exert on the car as the car
passes the highest point of the bump if the car travels at
a speed v? (b) What is the maximum speed the car can
have as it passes this highest point before losing contact
with the road?
Figure P6.46 Problems 46 and 47.
48. In one model of a hydrogen atom, the electron in orbit
around the proton experiences an attractive force of
about 8.20 � 10�8 N. If the radius of the orbit is 5.30 �
10�11 m, how many revolutions does the electron make
each second? (This number of revolutions per unit time
is called the frequency of the motion.) See the inside
front cover for additional data.
49. A student builds and calibrates an accelerometer, which
she uses to determine the speed of her car around a
certain unbanked highway curve. The accelerometer is
a plumb bob with a protractor that she attaches to the
roof of her car. A friend riding in the car with her observes
that the plumb bob hangs at an angle of 15.0°
from the vertical when the car has a speed of 23.0 m/s.
(a) What is the centripetal acceleration of the car
rounding the curve? (b) What is the radius of the
curve? (c) What is the speed of the car if the plumb bob
deflection is 9.00° while the car is rounding the same
curve?
50. Suppose the boxcar shown in Figure 6.13 is moving with
constant acceleration a up a hill that makes an angle �
with the horizontal. If the hanging pendulum makes a
constant angle � with the perpendicular to the ceiling,
what is a?
51. An air puck of mass 0.250 kg is tied to a string and allowed
to revolve in a circle of radius 1.00 m on a fric-
v