Planet Z is 7000 km in diameter The freefall acceleration on

Planet Z is 7000 km in diameter. The free-fall acceleration on the surface of Planet Z is 8.30 m/s2 . What is the escape velocity for something shot off the surface of Planet Z assuming that there is no resistance due to any atmosphere that might be present?

Solution

For any planet, the magnitude of gravitational acceleration is given by sqrt [GM/r^2]

Further, the escape velocity at any distance r from the from the centre of mass for the body is given by sqrt[2*G*M/r]

Using first relation, we have: 8.3 = sqrt [GM/r^2] ; That is, GM/r = 8.3*8.3*7000*10^3

Substituting the value of the above expression in the relation for excape velocity, we obtain

Ve = sqrt[2*8.3*8.3*7000*10^3] = 1.414*8.3*1000*2.646 = 31.05 x 10^3 m/s