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

To determine the minimum velocity needed for a spacecraft to reach the Moon's surface, we can apply the principle of conservation of energy.

First, let's consider the initial state of the spacecraft on the Earth's surface. We can assume that the spacecraft is at rest with respect to the Moon. At this point, the spacecraft has potential energy due to its position relative to the Moon.

Next, as the spacecraft travels from the Earth to the Moon, the force of gravity acts upon it. This force does work on the spacecraft, converting its potential energy into kinetic energy.

At the minimum velocity required for the spacecraft to reach the Moon's surface, all the initial potential energy is converted into kinetic energy. Any excess velocity would mean that the spacecraft would reach the Moon's surface with additional kinetic energy, requiring more energy to slow down or land safely.

The total energy of the spacecraft can be expressed as the sum of its potential energy (PE) and its kinetic energy (KE):

Total Energy = PE + KE

Since the spacecraft is initially stationary, its kinetic energy is zero (KE = 0). The total energy is equal to the potential energy:

Total Energy = PE

The potential energy can be calculated using the formula:

PE = m*g*h

Where:
m is the mass of the spacecraft,
g is the acceleration due to gravity, and
h is the distance between the Earth's surface and the Moon's surface.

Since we're neglecting the effect of air drag, the spacecraft does not lose energy due to any external factors. Therefore, the total energy at any point during the journey remains constant.

When the spacecraft reaches the Moon's surface, the height h is zero, resulting in the potential energy being zero as well. At this point, all the initial potential energy is converted into kinetic energy.

Now we can solve for the velocity needed to just make it to the Moon's surface. Using the conservation of energy principle:

Total Energy = PE + KE

Total Energy = 0 + 0 (since PE = 0 at the Moon's surface)

Total Energy = KE

The kinetic energy can be expressed as:

KE = (1/2) * m * v^2

Where:
v is the velocity of the spacecraft.

Setting the total energy equal to the kinetic energy, we get:

KE = (1/2) * m * v^2 = Total Energy

Since Total Energy = 0, we have:

(1/2) * m * v^2 = 0

Simplifying the equation, we find:

v^2 = 0

Taking the square root of both sides, we get:

v = 0

This implies that the minimum velocity needed for the spacecraft to just make it to the Moon's surface is zero. However, this is not practically possible as the spacecraft needs to travel at some finite velocity to overcome the Earth's gravitational pull and reach the Moon.

Therefore, the answer is that the spacecraft needs to travel at a velocity greater than zero to reach the Moon's surface.