## MYTUTOR SUBJECT ANSWERS

2433 views

### Derive the escape velocity from the surface of a planet with radius, r, and mass, M.

This question is about converting kinetic energy into gravitational potential energy. Escape velocity is the speed required to leave the gravitational field of a mass, in this case it's a planet. In other words the body has to be moving at such a velocity that it will reach a point infinitely far away from the planet.

The gravitational potential energy (PE) at any given point is given as:

PE = -GMm/r

Where G is Newton's gravitational constant; M is the mass of the planet; m is the mass of the body moving away from the planet and r is the distance from the centre of the planet/gravitational field.

Using this formula the potential energy at a distance infinitely far away (infinity) is 0. At the surface of the planet the potential energy is:

-GMm/r

This means that in order to get from the surface to infinity there will be a gain of

GMm/r

This will come from the kinetic energy of the body escaping.

Kinetic Energy (KE) is given as:

KE = mv2/2

Where m is the mass of the moving body and v is it's velocity.Now we have all we need to solve this problem

If we set the kinetic and potential energy equal to each other:

KE = PE

mv2/2 = GMm/r

Divide by m on both sides, this gets rid of all mentions of m. That means the final answer will not depend on the mass leaving the planet at all!

v2/2 = GM/r

Rearrange:

v = sqrt(2GM/r)

sqrt() means take the square root of what is inside the bracket.

Interesting related fact:

A black hole is an object that has an escape velocity that is greater than the speed of light. This means not even light can escape the gravitational pull of a black hole!!!

3 years ago

Answered by Thomas, who tutored A Level Physics with MyTutor

## Still stuck? Get one-to-one help from a personally interviewed subject specialist

#### 148 SUBJECT SPECIALISTS

£26 /hr

Degree: Environmental Geoscience (Masters) - Bristol University

Subjects offered:Physics, Maths+ 2 more

Physics
Maths
Geography
Biology

“Hello there! My name is Prina and I love studying and tutoring maths and science. Check out my profile for how I teach and previous students' feedback!”

£20 /hr

Degree: Physics (Bachelors) - Manchester University

Subjects offered:Physics, Maths+ 1 more

Physics
Maths
Chemistry

“I tutor Maths, Physics and Chemistry, I really enjoy my subjects, especially when I am tutoring others.”

MyTutor guarantee

|  1 completed tutorial

£20 /hr

Degree: Civil Engineering (Bachelors) - Manchester University

Subjects offered:Physics, Spanish+ 2 more

Physics
Spanish
Maths
Italian

“Hey! Do you want to learn stuff in a more interactive and funnier way? Then I'm the tutor you need!”

MyTutor guarantee

#### About the author

£20 /hr

Degree: Physics (Bachelors) - Oxford, Magdalen College University

Subjects offered:Physics, Maths+ 1 more

Physics
Maths
Further Mathematics

“Oxford physics student at Magdalen College. Here to help with Physics and Maths all the way up to A level.”

MyTutor guarantee

### You may also like...

#### Other A Level Physics questions

What's the highest height a ball thrown straight up will reach?

Is light the fastest? if no, then explain quantum entanglement!

What are the main observations of the alpha particle scattering experiment? (3)

A ball is thrown out of a window 18m off the ground. It is thrown horizontally at 0.5m/s. Show that it takes about 2s to reach the ground

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this.