Question

A 1.0kg stone and a 10.0kg stone are released from rest at the same height above the ground. Ignore air resistance. Which of these statements about the stones are true? Justify each answer. (a) Both have the same initial gravitational potential energy. (b) Both will have the same acceleration as they fall. (c) Both will have the same speed when they reach the ground. (d) Both will have the same kinetic energy when they reach the ground.

Short answer

(a) False

(b) True

(c) True

(D) False

Step-by-step solution

identification of given data.

• The mass of the stones is$1\mathrm{kg}$and$10\mathrm{kg}$.

Concept/Significance of Gravitational potential energy

The work done on a particle by a constant gravitational force is represented as a change in the gravitational potential energy which is given by,

${\mathrm{U}}_{\mathrm{grav}}=\mathrm{mgy}$

Here, m is the mass of an object, g is the acceleration due to gravity, and y is height.

Check whether the statement “Both have the same initial gravitational potential energy” is true or false(a)

The gravitational potential energy is given as$\mathrm{U}=\mathrm{mgy}$. From the expression of gravitational potential energy, it can be observed that the gravitational potential energy is directly proportional to the mass. So, the higher mass will have the higher potential energy.

Therefore, the given statement is false.

Check whether the statement “Both will have the same acceleration as they fall” is true or false(b)

The acceleration due to gravity has a constant value of$9.8\mathrm{m}/{\mathrm{s}}^2$on the surface of the earth. It does not depend upon the mass of the body in which gravitational force acts. So, both will have the same acceleration as they fall.

It is given by,

$\mathrm{g}=\frac{\mathrm{GM}}{{\mathrm{r}}^2}$

Here,$\mathrm{G}$is gravitational constant,$\mathrm{M}$is the mass of the earth, and$\mathrm{r}$its radius of the earth.

Therefore, the given statement is true.

Check whether the statement “Both will have the same speed when they reach the ground” is true or false(c)

The final speed is given by,

$\mathrm{v}=\sqrt{2\mathrm{gh}}$

From the above expression, it can be observed that the speed depends upon the height from which the object is released and the acceleration due to gravity. It does not depend upon the mass of the object. So, the speed for both objects will be the same.

Therefore, the given statement is true.

Check whether the statement “Both will have the same kinetic energy when they reach the ground” is true or false(d)

The expression of kinetic energy is given by,

$\mathrm{K}=\frac{1}{2{\mathrm{mv}}^2}$

From the above expression, it can be observed that the kinetic energy of bodies is depending upon the mass of the body. So, the higher mass will have the higher kinetic energy.

Therefore, the given statement is false.