Escape velocity of an Earth-Moon system

[cragar]

Homework Statement

What velocity is required to escape from the Earth-Moon system from the surface of
the Moon? Assume that all of the necessary velocity is imparted at once, as with a
cannon or rail gun on the Moon itself. In what direction must the initial velocity vector
be pointed to ensure the lowest escape velocity?

The Attempt at a Solution

Would i just find the center of mass between Earth and the moon and then use the distance from the center of mass to the surface of the moon to find the escape velocity.

 

[BvU]

What you need is some relevant equations. Any suggestions ?
Suppose you have this center of mass position. Then what ?

 

[tjmiller88]

Ask yourself: what is escape velocity?

What are you "escaping" from and how might it be relevant to the Earth and Moon?

Your starting point, once you've answered those questions, should be a diagram of the Earth and Moon, showing some relevant forces.

 

[gneill]

Ask yourself: what is escape velocity?

What are you "escaping" from and how might it be relevant to the Earth and Moon?

Your starting point, once you've answered those questions, should be a diagram of the Earth and Moon, showing some relevant forces.

...and velocities; The Earth and Moon are not stationary with respect to one another

 

[HalfLostAndSearching]

I would like to clarify that the concept of escape velocity applies to any celestial body, not just the Earth-Moon system. However, for the purpose of this question, I will focus on the Earth-Moon system.

To calculate the escape velocity from the surface of the Moon, we can use the formula:

Ve = √(2GM/r)

Where Ve is the escape velocity, G is the gravitational constant, M is the mass of the Moon, and r is the distance from the center of the Moon to its surface.

Plugging in the values for the Earth-Moon system, we get an escape velocity of approximately 2.38 km/s. This means that any object launched from the surface of the Moon with a velocity of 2.38 km/s or higher will escape the Moon's gravitational pull and enter into orbit around the Earth.

To answer the second part of the question, the initial velocity vector must be pointed in the opposite direction of the Moon's gravitational pull, which is towards the Earth. This means that the initial velocity must have a vertical component to overcome the gravitational force and a horizontal component to provide the necessary tangential velocity for orbit.

It is important to note that the escape velocity will vary depending on the initial location and direction of the object. For example, an object launched from a higher altitude on the Moon's surface will have a lower escape velocity compared to an object launched from a lower altitude, as it has a head start in overcoming the Moon's gravitational pull.

I hope this explanation helps in understanding the concept of escape velocity in the Earth-Moon system.