91Ó°ÊÓ

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

$$\begin{gathered} {\mathit{W}}_{\mathbf{\text{grav}}}\mathbf{=}\mathit{Δ}{\mathit{U}}_{\mathbf{\text{grav}}} \\ \mathbf{=}{\mathit{U}}_{\mathbf{\text{grav}}\mathbf{,}\mathbf{1}}\mathbf{-}{\mathit{U}}_{\mathbf{\text{grav}}\mathbf{,}\mathbf{2}} \\ \mathbf{=}\mathit{m}\mathit{g}{\mathit{y}}_{\mathbf{1}}\mathbf{-}\mathit{m}\mathit{g}{\mathit{y}}_{\mathbf{2}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right) \end{gathered}$$

Here, assume that m is the mass of the egg ${\mathit{y}}_{\mathbf{1}}$and ${\mathit{y}}_{\mathbf{2}}$are the initial and final height of the building, g is the acceleration due to gravity, and localid="1668142626159" ${\mathit{W}}_{\mathbf{\text{grav}}}$, is the change in the gravitational potential energy.

The student on the roof will assign a coordinate of to the initial position of the egg and a final position of $-h$, here is the height of the roof when measured from the ground.

On the other hand, the student on the ground will assign a coordinate of to the initial position and a coordinate of to the final position.

Case 1:

When the person is on the roof.

Substitute $-h$for$y_1$ , and $0$for$y_2$ in equation (1).

$$\begin{gathered} W_{\text{grav}}=mg{y}_1-mg{y}_2 \\ =-mgh-0 \\ =-mgh \end{gathered}$$

Case 2:

When the person is on the ground.

Substitute 0 for $y_1$, and $h$for $y_2$in equation (1), and we get,

$$\begin{gathered} W_{\text{grav}}=mg{y}_1-mg{y}_2 \\ =0-mgh \\ =-mgh \end{gathered}$$

Therefore, in both the cases the initial and final gravitational potential energies are different, but the change in gravitational potential energy and the kinetic energy of the egg just before it strikes the ground are the same.