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.

To calculate the minimum velocity needed for the spacecraft to just make it to the Moon's surface, we'll use the concept of orbital mechanics.

First, we need to understand the initial conditions for the spacecraft's launch:

1. The spacecraft is launched from the Earth's surface.
2. The spacecraft reaches a point where it is affected by the Moon's gravity.
3. The spacecraft's velocity needs to be precisely balanced such that it enters the Moon's gravitational field without crashing into it or escaping it.

To solve this problem, we can use the concept of the circular velocity and gravitational force.

1. Circular Velocity: The velocity needed for an object to stay in a circular orbit around a celestial body is given by the formula:

v = sqrt(G * M / r)

Where:
- v is the circular velocity
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
- M is the mass of the celestial body (in this case, the Moon)
- r is the distance between the spacecraft and the center of the Moon

2. Gravitational Force: The gravitational force acting on the spacecraft is given by:

F = (G * m * M) / r^2

Where:
- F is the gravitational force
- m is the mass of the spacecraft

To just make it to the Moon's surface, we want to balance these two forces. The circular velocity should be equal to the velocity of the spacecraft when it reaches the Moon (just enough to counteract the gravitational pull of the Moon).

Now, here's how to calculate the minimum velocity:

1. Determine the distance between the Earth and the Moon. This is approximately 384,400 kilometers.

2. Calculate the mass of the Moon. This is approximately 7.35 x 10^22 kilograms.

3. Assume a reasonable altitude from the Moon's surface where the spacecraft will land. Let's say 100 kilometers.

4. Add the radius of the Moon (1,737 kilometers) to the altitude to get the distance (r) from the center of the Moon to the spacecraft.

5. Use the formula for the circular velocity to calculate the minimum velocity needed:

v_min = sqrt(G * M / r)

6. Calculate the minimum velocity based on the known values:

v_min = sqrt(6.67430 × 10^-11 * 7.35 x 10^22 / (1.737 x 10^6 + 100) * 10^3)

Convert the distance from kilometers to meters.

7. Solve the equation to find v_min.

The minimum velocity needed for the spacecraft to just make it to the Moon's surface is the result.

Note: This calculation ignores air drag from the Earth's atmosphere, which is reasonable for space missions.