The first U.S. spacecraft to photograph the Moon close up was the unmanned
Ranger 7 photographic mission in 1964. The spacecraft, shown in the NASA photograph below, contained television cameras that transmitted close-up pictures of the Moon back to Earth as the spacecraft approached the Moon. The spacecraft did not have retro-rockets to slow itself down, and it eventually simply crashed onto the Moon's surface, transmitting its last photos immediately before impact.
The next figure is the first image of the Moon taken by a U.S. spacecraft, the
Ranger 7, on July 31, 1964, about 17 minutes before impact on the lunar surface. The large crater at center right is Alphonsus
(108 km diameter); above it (and to the right) is Ptolemaeus and below it is Arzachel. The
Ranger 7 impact site is off the frame, to the left of the upper left corner. To find out more about the actual
Ranger lunar missions,
click here.
To send a spacecraft to the Moon, we put it on top of a large rocket containing lots of rocket fuel and fire it upward. At first the huge ship moves quite slowly, but the speed increases rapidly. When the "first-stage" portion of the rocket has exhausted its fuel and is empty, it is discarded and falls back to Earth. By discarding an empty rocket stage we decrease the amount of mass that must be accelerated to even higher speeds. There may be several stages that operate for a while and then are discarded before the spacecraft has risen above most of Earth's atmosphere (about 50 km, say, above the Earth), and has acquired a high speed. At that point all the fuel available for this mission has been used up, and the spacecraft simply coasts toward the Moon through the vacuum of space.
You will model the
Ranger 7 mission. Starting 50 km above the Earth's surface
(5 × 104 m),
a spacecraft coasts toward the Moon with an initial speed of about
1 × 104 m/s.
Here is the data you should use:
G = 6.7  10-11 N · m2/kg2; |
| mass of spacecraft = 173 kg; |
| mass of Earth ≈ 6 1024 kg; |
| mass of Moon ≈ 7 1022 kg; |
| radius of Earth = 6.4 106 m; |
| radius of Moon = 1.75 106 m; |
| distance from Earth to Moon ≈ 4 108 m (400,000 km, center to center). |
We're going to ignore the Sun in a simplified model even though it exerts a sizable gravitational force. We're expecting the Moon mission to take only a few days, during which time the Earth (and Moon) move in a nearly straight line with respect to the Sun, because it takes 365 days to go all the way around the Sun. We take a reference frame fixed to the Earth as representing (approximately) an inertial frame of reference with respect to which you can use the Momentum Principle.
For a simple model, let the Earth and Moon be fixed in space during the mission. Factors that would certainly influence the path of the spacecraft include the motion of the Moon around the Earth and the motion of the Earth around the Sun. In addition, the Sun and other planets exert gravitational forces on the spacecraft. As a separate project you might like to include some of these additional factors.
(a) Compute the path of the spacecraft, and display it either with a graph or with an animated image. (Submit a file with a maximum size of
1 MB. Your program should end with ".py".)
In parts (b) and (c), report the step size
Δt that gives accurate results (that is, cutting this step size has little effect on the results.)
(b) By trying various initial speeds, determine the
minimum launch speed needed to reach the Moon, to three significant figures (this is the speed that the spacecraft obtains from the multistage rocket, at the time of release above the Earth's atmosphere, which we'll take to be
50 km = 5 × 104 m
above the Earth's surface.)
minimum launch speed
m/s
What happens if the launch speed is less than this minimum value? (To be sure that inaccuracies due to time steps do not affect your answer, consider launch speeds which are 90% or less of the minimum launch speed.)
Approximately, what is the largest step size
Δt you can use without significantly changing the three-significant-figure result for part (a)?
s
(c) Use a launch speed 10% larger than the minimum value found in part (a). How long does it take to go to the Moon, in days, to two significant figures? (Be sure to check the step size issue.)
(d) What is the "impact speed" of the spacecraft (its speed just before it hits the Moon's surface)? Make sure that your spacecraft crashes on the surface of the Moon, not at the Moon's center!
m/s
You may have noticed that you don't actually need to know the mass
m of the spacecraft in order to carry out the computation. The gravitational force is proportional to
m, and the momentum is also proportional to
m, so
m cancels. However, nongravitational forces such as electric forces are not proportional to the mass, and there is no cancellation in that case. We kept the mass
m in the analysis in order to illustrate a general technique for predicting motion, no matter what kind of force, gravitational or not.