Question

Can you think of a scenario when the kinetic and gravitational potential energy could both be zero ? Describe or draw how this could be possible below

Answer 1

Both kinetic and gravitational potential energy can become zero at infinite distance from the Earth.

Consider an object  of mass m projected from the surface of the Earth with a velocity v.

The total energy of the body on the surface of the Earth is the sum of its kinetic energy $\frac{1}{2} mv^2$and gravitational potential energy $-\frac{GMm}{R^2}$.

here, M is the mass of the Earth, R is the radius of Earth and G is the universal gravitational constant.

The gravitational potential energy of the object is negative since it is in an attractive field, which is the gravitational field of the Earth.

The energy of the object on the surface of the earth is given by,

$E_i=\frac{1}{2} mv^2-\frac{GMm}{R^2}$

As the object rises upwards, it experiences deceleration due to the gravitational force of the Earth. Its velocity decreases and hence its kinetic energy decreases.

The decrease in kinetic energy is manifested as  an equal increase in potential energy. The potential energy becomes less and less negative as more and more kinetic energy is converted into potential energy.

At a height h from the surface of the Earth, the energy of the object is given by,

$E_h=\frac{1}{2} mv_h^2-\frac{GMm}{(R+h)^2}$

The velocity $v_h$ is less than v.

When h =∞, the gravitational potential energy increases from a negative value to zero.

If the velocity of projection is adjusted in such a manner that the velocity decreases to zero at infinite distance from the earth, the object's kinetic energy also becomes equal to zero.

Thus, it is possible for both kinetic and potential energies to be zero at infinite distance from the Earth. In this case, kinetic energy decreases from a positive value to zero and the gravitational potential energy increases from  a negative value to zero.