Question

In 1971 Alan Shepard and Ed Mitchell spent
33 hours on the Moon as crew members of
Apollo 14. Shepard, one of the original
Mercury astronauts, was 47 years old and is
the oldest man to walk on the lunar surface.
While on the Moon, Shepard hit two golf balls
using an improvised golf club. After striking
the second ball, Shepard said “miles and miles
and miles”, a reference to the distance the
ball travelled. Owing to the lack of an atmosphere and the low acceleration due to gravity on the Moon,
a golf ball should travel a great distance compared to a golf ball hit on the Earth. So, how far should a
golf ball travel on the Moon? We can obtain a pretty good estimate if we make some good, reasonable
assumptions and use our knowledge of physics. Does a golf ball go miles on the Moon? Let’s find out.
a. Write the kinematic equations for the motion of the golf ball on the Moon*. Write separate
equations for the x-axis and z-axis (see diagram). Remember, there aren’t any aerodynamic
forces acting on the ball on the Moon. The trajectory and, therefore, the distance travelled
will be functions of V0,  and glunar.
b. You will now have two equations: z = f(glunar, V0, , t) and x = f(V0, , t). Use the equation for z
to eliminate t from the x equation. (Hint: set z = 0 and solve for t). Now you’ll have an
equation for x= f(V0, glunar). For a given, fixed value of V0, find the value of  that
maximizes x. You have to use your knowledge of differential calculus to do this. Remember
how to find the maxima and the minima of a function? You must find  that maximizes x
analytically. Show all work.
c. Okay, you have a value of  that maximizes x. Let’s approximate a value for V0. Let’s start
with a high value of V0 to get an upper bound. The V0 for professional golfers averages about
167 mph. For amateur golfers the average V0 is about 133 mph. Alan Shepard had to swing
his “club” with one arm wearing an extremely bulky space suit. His flexibility was very
limited. Let’s say his ball’s V0 was half that of an amateur’s on Earth; so V0 = 67 mph. Since
this is an estimate, let’s call it 70 mph. What is xmax for this V0?
d. Now assume that somehow (magically) Shepard was able to achieve a V0 = 167 mph. What is
xmax for this V0? Did it go anywhere close to a mile?
e. What was the apogee of Shepard’s golf ball? Use calculus and show your calculations.

Answer 1