Chapter 3, Problem 84A

(a)

To determine

To Compare: The maximum height of the ball achieved on Jupiter with that on the Earth.

Explanation of Solution

Given:

Ball is thrown on Jupiter and Earth with the same velocity.

The free-fall acceleration on Jupiter is three times of that on the Earth.

Formula Used:

Third equation of motion:

  $v^2-u^2=2as$

Where, v is the final velocity, u is the initial velocity, a is the acceleration and s is the displacement.

Calculations:

The final velocity at the maximum height is zero, $v=0$ .

The initial velocity is same. $u_E=u{J}_{}=u$

Acceleration due to gravity on Jupiter, $g_J=3g_E$

Where, $g_E$ is the acceleration due to gravity on Earth.

Let the height achieved on Jupiter be $h_J$ and that on Earth be $h_E$ .

For Jupiter:

  $\begin{array}{l}0-u_J{}^2=-2g_J{h}_J\\ \Rightarrow h_J=\frac{u^2}{2g_J}\end{array}$

For Earth:

  $\begin{array}{l}0-u_E{}^2=-2g_E{h}_E\\ \Rightarrow h_E=\frac{u^2}{2g_E}\end{array}$

Take the ratio:

  $\begin{array}{l}\frac{h_J}{h_E}=\frac{g_E}{g_J}\\ \Rightarrow h_J=\frac{1}{3}{h}_E\end{array}$

Conclusion:

Thus, the height achieved by ball on Jupiter would be one-third to that on the Earth.

(b)

To determine

To Compare: The maximum height of the ball achieved on Jupiter with that on the Earth.

Explanation of Solution

Given:

Ball is thrown on Jupiter with three times more velocity than that on the Earth.

The free-fall acceleration on Jupiter is three times of that on the Earth.

Formula Used:

Third equation of motion:

  $v^2-u^2=2as$

Where, v is the final velocity, u is the initial velocity, a is the acceleration and s is the displacement.

Calculations:

The final velocity at the maximum height is zero, $v=0$ .

The initial velocity on Jupiter, $u{J}_{}=3u_E$

Acceleration due to gravity on Jupiter, $g_J=3g_E$

Where, $g_E$ is the acceleration due to gravity on Earth.

Let the height achieved on Jupiter be $h_J$ and that on Earth be $h_E$ .

For Jupiter:

  $\begin{array}{l}0-u_J{}^2=-2g_J{h}_J\\ \Rightarrow h_J=\frac{u_J{}^2}{2g_J}\end{array}$

For Earth:

  $\begin{array}{l}0-u_E{}^2=-2g_E{h}_E\\ \Rightarrow h_E=\frac{u_E{}^2}{2g_E}\end{array}$

Take the ratio:

  $\begin{array}{l}\frac{h_J}{h_E}=\frac{u_J{}^2{g}_E}{u_E{}^2{g}_J}\\ \Rightarrow h_J=\frac{{\left(3\right)}^2}{3}{h}_E\\ \Rightarrow h_J=3h_E\end{array}$

Conclusion:

Thus, the height achieved by ball on Jupiter would be three times to that on the Earth if the initial velocity is three times.