# Calculate Moon's Potential/Kinetic Energy, Escape Velocity & Period

• EthanVandals
In summary: F = (GMm)/r^2. If you're looking for a specific formula for a specific situation, there is a more specific formula for that.However, for the purpose of this problem, the approximation given is adequate.
EthanVandals

## Homework Statement

Calculate the potential energy and kinetic energy of the moon, as well as it's escape velocity. Give the moon's period and angular speed. Consider the Earth to have a mass of 6 x 10^24 kg, the moon to have a mass of 7 x 10^22 kg, and their center separation distance to be 4 x 10^8 m.

## Homework Equations

I = mr^2
F(g) = (GMm)/r^2
KE = 1/2(I)(Omega)^2
PE = ?
Omega = ChangeinTheta/ChangeinTime

## The Attempt at a Solution

So far, I have drawn my free body diagram, and I've calculated the force due to gravity (the centripetal force in this case) to be 1.750875 x 10^20. This is based off the the F(g) formula. From here, I don't know where to move, because to calculate the other items I need, I believe I need the velocity of the moon, which is not provided. Can anyone give me a hand with this? Thanks!

EthanVandals said:
1 I believe I need the velocity of the moon, which is not provided. Can anyone give me a hand with this?
How is centripetal force related to speed? (Review your lessons on uniform circular motion.)

Also, you have a question mark for the formula for gravitational potential energy. This is a basic formula which you should be able to easily find in your textbook or online.

Also, you know (or can look up) how long it takes the moon to make one orbit, and you know the diameter of the orbit, so it is a simple matter to calculate the moon's speed.

TSny said:
How is centripetal force related to speed? (Review your lessons on uniform circular motion.)

Also, you have a question mark for the formula for gravitational potential energy. This is a basic formula which you should be able to easily find in your textbook or online.
I mean, for PE, the only thing that shows up is mgh when I search it online. I also know that Centripetal Force and speed are related in the fact that the object must be moving at a certain speed, or else it would get pulled to the center, or it would go flying away and leave orbit. I found an equation that stated Fc = (mv^2)/r. I solved for v in this case, and I got a velocity of approximately 1000 m/s. I plugged in this value for the v in the omega = v/r equation and got a value of omega as 2.500624923 x 10^-6. I then plugged the mass and the radius into the equation Inertia = mass(radius)^2, and got a value of 1.12 x 10^40 for my inertia. I plugged both of these values into the equation KE=(1/2)(inertia)(omega)^2 and got a kinetic energy of 3.921960003 x 10^68. Is all of that correct so far?

EthanVandals said:
I don't know where to move, because to calculate the other items I need, I believe I need the velocity of the moon, which is not provided.

You know the radius so you know the circumference of the orbit and it's not hard to find out how long an orbit takes.

CWatters said:
You know the radius so you know the circumference of the orbit and it's not hard to find out how long an orbit takes.

EthanVandals said:
I found an equation that stated Fc = (mv^2)/r. I solved for v in this case,

Actually I think your method is better as it's only based on data in the problem statement.

You should work with units.
EthanVandals said:
I mean, for PE, the only thing that shows up is mgh when I search it online.
That is only an approximation for small heights (e.g. in a building). There is a more general formula based on F = (GMm)/r^2.

EthanVandals said:
and got a value of 1.12 x 10^40 for my inertia.
Okay so far, apart from the missing units.
EthanVandals said:
I plugged both of these values into the equation KE=(1/2)(inertia)(omega)^2 and got a kinetic energy of 3.921960003 x 10^68
You multiply something with 1040 with numbers smaller than 1. How can the result be larger?

You have the mass of the Moon and its speed, there is no need to use the moment of inertia as detour.

## 1. What is the formula for calculating the Moon's potential energy?

The formula for calculating the Moon's potential energy is PE = mgh, where m is the mass of the Moon, g is the acceleration due to gravity, and h is the height of the Moon from the surface of the Earth.

## 2. How do you calculate the Moon's kinetic energy?

The formula for calculating the Moon's kinetic energy is KE = 1/2mv^2, where m is the mass of the Moon and v is its velocity.

## 3. What is the escape velocity of the Moon?

The escape velocity of the Moon is approximately 2.38 km/s. This means that an object needs to be moving at a speed of at least 2.38 km/s to escape the Moon's gravitational pull.

## 4. How do you calculate the period of the Moon's orbit around the Earth?

The formula for calculating the period of the Moon's orbit is T = 2π√(a^3/GM), where a is the distance between the Moon and the Earth, G is the gravitational constant, and M is the mass of the Earth.

## 5. What is the relationship between potential energy and kinetic energy in the Moon's orbit?

The potential energy and kinetic energy of the Moon in its orbit are constantly changing, but their sum remains constant. As the Moon moves closer to the Earth, its potential energy decreases and its kinetic energy increases. As it moves further away, the opposite occurs.

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