Forces and Motion

Equation

v = (2GM/r)½

1.17

Orbital Motion

Escape Velocity

The velocity it takes to completely escape the pull of Earth’s gravity (or the gravitational pull of any object) is called the “escape velocity”. If a spacecraft is heading for the Moon, Mars or any other planet it needs to reach escape velocity, which is the point at which the upward kinetic energy (1/2 mv2) is just greater than the downward potential energy due to gravity. The escape velocity for the Earth is 11.2 km/s. We can calculate the escape velocity for a spherical body by setting the kinetic energy equal to the gravitational potential energy.

½ mv2 = GMm/r

Solving for v, we get:  v = (2GM/r)½

Here again, the escape velocity is independent of the mass of the satellite or rocket and only depends on the mass of the central body. Using parameters for Earth and the kg-m-s units, we find the escape velocity for Earth to be:

v = [(2 x 6.67 x 10-11 x 5.97 x 1024) / 6.378 x 106]½= 11,174 m/s = 11.2 km/s

Try This!

Add another column to the table above where you computed orbital velocities and add escape velocity. What is the escape velocity of a basketball? A neutron star? Take care to ensure you are using the correct units for distance and mass. You may need to convert from meters to kilometers.

 Object Mass (kg) Radius (km) Orbital Radius (km) Orbital Velocity (km/s) Moon 7.36 x 1022 1,737 1,937 Mars 6.42 x 1023 3,397 3,597 Sun 2.00 x 1030 695,500 149.6 Neutron Star 2.8 x 1030 20 220 Milky Way Galaxy 1.16 x 1042 2.55 x 1017 2.55 x 1017 Basketball 0.5 4.7 x 10-5 0.001

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