Question

At some planet 'g' is 1.96 $m/{s^2}$ . If it is safe to jump from a height of 2m on earth, then what should be the corresponding safe height for jumping on that planet?

Answer 1

In this question, we will derive the relation between acceleration due to gravity and height. Then by using this relation we will further solve the question by putting the value of gravity we can get the height. Further, we will study the basics of a solar system and also we will see the structure of our Earth, for better understanding.
Formula used:
$\dfrac{{{g_h}}}{g} = \left( {1 - \dfrac{{2h}}{R}} \right)$

Complete step-by-step solution
As we know that, potential energy is defined as the energy that is stored in an object due to its position relative to some zero position. Also, an object or a body possesses gravitational potential energy when it is positioned or placed at a height above or below the zero height.
Here, we will first derive the relation between the acceleration due to gravity and height.
Let us know the acceleration due to gravity on earth is given by:

& g=\dfrac{GM}{{{R}^{2}}} \\\ & \Rightarrow GM=g{{R}^{2}}.....(1) \\\ \end{aligned}$$ Now we could get the acceleration due to gravity at a height h from the surface of our earth as, $$\begin{aligned} & {{g}_{h}}=\dfrac{GM}{{{(R+h)}^{2}}} \\\ & \Rightarrow GM={{(R+h)}^{2}}{{g}_{h}}........(2) \\\ \end{aligned}$$ Now, by using the obtained equations (1) and (2), we get, $$\dfrac{{{g}_{h}}}{g}=\dfrac{{{R}^{2}}}{{{(R+h)}^{2}}}$$ $$\therefore \dfrac{{{g}_{h}}}{g}=\left( 1-\dfrac{2h}{R} \right)............(3)$$ So, we get the relation between the acceleration due to gravity and the height. Now, we have the value of acceleration due to gravity as: $${{g}_{e}}=9.8m/{{s}^{2}}$$ Also, we know the height is given by: $$h=2m$$ At some other planet, acceleration due to gravity at earth is given by: $${{g}_{p}}=1.96m/{{s}^{2}}$$ Taking ratio of the above two values of acceleration due to gravity and putting the values in equation (3), we get: $$\dfrac{{{g}_{p}}}{{{g}_{e}}}=\dfrac{1}{5}$$ From this, we can observe that at some planet the value of g decreases by 5. So, the height will increase 5 times. $$\begin{aligned} & h=5\times 2 \\\ & \therefore h=10m \\\ \end{aligned}$$ **Therefore, the safe height to jump on the planet is given by the above equation.** **Note:** Here, one should note that acceleration due to gravity varies inversely with height or altitude. Also, if a particle or body is in motion but does not have any height, then the potential energy of the particle of the body in that case will be zero. In the solution, we have just made a comparison between the respective acceleration due to gravity of the planet and earth.