10.2b - Escape Velocity

Escape speed (vesc) the speed an object needs to have at the surface of a planet in order to have enough kinetic energy to escape the planet’s gravitational pull. For an object at orbital radius r (m) about a planet of mass M (kg), the escape speed in m s–1 is given by vesc=2GMr, where G is the universal gravitational constant.

Orbital speed (vorbit) the linear orbital speed of a satellite about a planet. At an orbital radius r (m) above a planet of mass M (kg), vorbit (m s–1) is given by vorbit=GMr, where G is the universal gravitational constant.

1. A planet is in a circular orbit around a star. The speed of the planet is constant. The following data are given:

Mass of planet = 8.0×10ˆ24 kg

Mass of star = 3.2×10ˆ30 kg

Distance from the star to the planet R  = 4.4×10ˆ10 m.

A spacecraft is to be launched from the surface of the planet to escape from the star system. The radius of the planet is 9.1 × 103 km.

1. Explain why a centripetal force is needed for the planet to be in a circular orbit. [2]

2. Calculate the value of the centripetal force. [1]

3. Show that the gravitational potential due to the planet and the star at the surface of the planet is about −5 × 109 J kg−1. [3]

2. Calculate a body’s escape velocity from the moon. Assume that the moon is a uniform sphere with a radius of 1.76×10^6 m and a mass of 7.36×10^22 kg.

3. A black hole is a body from which nothing can escape. What conditions must be met for a uniform spherical body of mass M to be a black hole? What should the radius of a black hole be if it has a mass nine times that of the Earth? 

Mass of Earth = M_E = 6x 10^24 kg

Speed of Light = c = 3.00 x 10^8 m/s