# Escape Velocity

This topic is part of the HSC Physics syllabus under the section Motion in Gravitational Fields.

### HSC Physics Syllabus

• Derive quantitatively and apply the concepts of gravitational force and gravitational potential energy in radial gravitational fields to a variety of situations, including but not limited to:
– the concept of escape velocity v_(esc)=sqrt((2GM)/r)
– total potential energy of a planet or satellite in its orbit U=-(GMm)/r

### What is Escape Velocity?

Escape velocity is a fundamental concept in astrophysics and space exploration. It refers to the minimum speed needed for an object to break free from the gravitational field of a celestial body, like a planet or moon, without further propulsion.

### Derivation of Escape Velocity Formula

To escape the gravitational field of a celestial body, an object must overcome the pull of gravity exerted by that body. This requires the object to have sufficient kinetic energy to counteract the gravitational potential energy holding it in place.

Imagine an object travelling at escape velocity, eventually reaching faraway enough to finally escape the effect of gravity. By Newton’s universal law of gravitation, this scenario can only occur at r = oo since:

$$g = \frac{GM}{r^2}$$

• As a mass travels away from the centre of a gravitational field, its kinetic energy is transformed to gravitational potential energy. Therefore, kinetic energy decreases while gravitational potential energy increases.
• At r = oo both gravitational potential and kinetic energy of the object both become zero. Therefore, the final total energy also becomes zero.
• By using law of conservation of energy, the initial total energy must be equal to the final total energy which is zero:

K+U=0

1/2mv^2=(GMm)/r

v^2=(2GM)/r

v_(esc)=sqrt((2GM)/r)

We can use this equation to determine the minimal velocity required to allow an object to escape the gravitational field it is in.

Like orbital velocity, escape velocity is independent of the mass of the object that tries to escape the gravitational field. Escape velocity is only dependent on the mass M whose gravitational field the object is escaping from, and the distance of the object from the centre of this mass, r.

### Escape Velocity vs Orbital Velocity

• Do not confused this equation with orbital velocity – they are fundamentally very different concepts.
• Escape velocity is always greater than orbital velocity at the same position in a gravitational field because more kinetic energy is needed to ‘overcome’ the gravitational force than to accompany it.
• When an object's velocity increases to and beyond its escape velocity while in orbit, it will leave this orbit and continue moving away from the object that it was originally orbiting around.

### Calculation Example

Calculate the escape velocity of a 70 kg person on the surface of Earth.

Solution:

Note it is essential for you to provide the derivation (see above) of the escape velocity formula for calculations in assessments as it is not listed on the NESA HSC Physics formula sheet.

v_(esc) = sqrt((2GM)/r)

v_(esc) = sqrt((2(6.67 xx 10^{-11})(6.0 xx 10^{24}))/(6.371 xx 10^6))

v_(esc) = 11200 \text{ m/s}

Previous section: Gravitational Potential Energy

Next section: Energy Changes in and between Orbits