Question

A man can jump to a height of $1.5m$ on a planet A what is the height he may be able to jump on another planet whose density and radius are respectively $one-quarter$ and $one-third$ that of planet A

Answer 1

Step 1: Given data

Maximum height of jump on a planet A, $H_{max}=1.5m$

Step 2: Find the relation between $\mathit{g}$ (gravity due to earth) and $\mathit{g}\mathbf{\text{'}}$ (gravity due to another planet)

Gravity due to earth, $g=GM/r^2$

$=\frac{G\times \rho \times \frac{4}{3}\times \pi \times r^3}{r^2}$

$=G\times \rho \times \frac{4}{3}\times \pi \times r^{}$

Gravity due to another planet, $g^{\text{'}}=GM/{\left(r^{\text{'}}\right)}^2$

$$\begin{gathered} =\frac{G\times \frac{\rho }{4}\times \frac{4}{3}\times \pi \times {\left(\frac{r}{3}\right)}^3}{{\left(\frac{r}{3}\right)}^2} \\ =G\times \frac{\rho }{4}\times \frac{4}{3}\times \pi \times \left(\frac{r}{3}\right) \end{gathered}$$

$$\begin{gathered} \frac{g}{g^{\text{'}}}=\frac{G\times \rho \times \frac{4}{3}\times \pi \times r}{G\times {\displaystyle \frac{\rho }{4}}\times \frac{4}{3}\times \pi \times \left(\frac{r}{3}\right)} \\ \frac{g}{g\text{'}}=12 \end{gathered}$$

Step 3: Find the maximum height of jump at another planet, $\mathit{H}{\mathbf{\text{'}}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$

As, $H_{max}\propto \frac{u^2}{2g}$

$$\begin{gathered} \frac{H{\text{'}}_{max}}{H_{max}}=\frac{g}{g\text{'}} \\ H{\text{'}}_{max}=12\times 1.5m \\ H{\text{'}}_{max}=18m \end{gathered}$$

Hence, height which a person can jump on another planet is$18m$.