Q4DQ - University Physics with Modern Physics

Question

An egg is released from rest from the roof of a building and falls to the ground. Its fall is observed by a student on the roof of the building, who uses coordinates with origin at the roof, and by a student on the ground, who uses coordinates with origin at the ground. Do the values the two students assign to the following quantities match each other: initial gravitational potential energy, final gravitational potential energy, change in gravitational potential energy, and kinetic energy of the egg just before it strikes the ground? Explain.

 

Step-by-Step Solution

Answer

The initial and final gravitational potential energies are different, but the change in gravitational potential energy and the kinetic energy of the egg is the same

 

Step 1: Concept/Significance of Work done

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,

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

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  ${\mathit{W}}_{\mathbf{\text{grav}}}$, is the change in the gravitational potential energy.

Step 2: Determine whether the values the two students assigned to the following quantities match each other

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.