To project a body of mass m from earth's surface to infinity, the required kinetic energy is (assume, the radius of earth is R_E, g= acceleration due to gravity on the surface of earth):

Solution

Physics sir · 4 steps

Give the body just enough kinetic energy to cancel its negative potential energy at the surface.

1. Energy at the surface. On the surface the body has potential energy U = -GMm/R_E. At infinity its potential energy is zero, and with the least energy it arrives there with zero speed.
2. Conserve energy. The total energy at the surface must equal the total energy at infinity, which is zero. So the kinetic energy needed is K = GMm/R_E.

   K = GMm/R_E = GM/R_E² m R_E = m g R_E

3. Use g. On the surface g = GM/R_E², so GM = gR_E². Putting this in gives K = mgR_E. This matches 1/2mvₑ² with vₑ = √2gR_E.
4. So. The required kinetic energy is mgR_E, option (c).

The idea to keep. Escape energy is GMm/R. Swap GM for gR² and it becomes mgR. Escape speed is √2gR because the 2 comes from the 1/2 in kinetic energy.

Answer from the NTA final answer key. Solution by PrepWiser.