physics
posted by Anonymous .
Consider a spacecraft that is to be launched from the Earth
to the Moon. Calculate the minimum velocity needed for the
spacecraft to just make it to the Moon’s surface. Ignore air drag
from the Earth’s atmosphere. Hint: The spacecraft will not have
zero velocity when it reaches the Moon

Contrary to popular belief, a spacecraft does not have to reach full escape velocity in order to reach the Moon. The full escape velocity from a 200 mile high circular orbit is 24,400 mph. Assuming the trip starts from a 200 mile high circular orbit, the minimum injection velocity out of this orbit would be ~24,200 miles per hour, ~35,500 feet per second, or ~10.82 km/sec., to place the spacecraft on an elliptical, minimum energy, Hohmann Transfer trajectory to the Moon. Since the spacecraft is already traveling at a speed of 25,306 ft/sec to maintain the 200 mile high orbit, the deltaV, additional velocity, needed out of the circular orbit is ~10,194 ft/sec. This elliptical trajectory, eccentricity = .966, would bring the spacecraft tangent to the lunar orbit in 120 hours. Any less velocity and a spaceship would not get there at all. Any more velocity and the time would be shortened as well as the spacecraft passing in front of the Moon. The Apollo missions typically took about 72 hours to reach the moon. As the spacecraft passed in front of the Moon, the Service Module rocket engine fired, slowing the spacecraft down to a velocity that placed the spacecraft into lunar orbit.