The ratio of radii of earth to another planet is `2//3` and the ratio of their mean densities is `4//5`. If an astronaut can jump to a maximum height of `1.5 m` on the earth, with the same effort, the maximum height he can jump on the planet is

Options

A. 1 m
B. 0.8 m
C. 0.5 m
D. 1.25 m

Answer

To solve the problem, we need to find the maximum height an astronaut can jump on another planet given the ratios of the radii and densities of Earth and that planet. We will use the relationship between gravitational acceleration and the parameters given.

### Step-by-Step Solution:

1. **Identify the Given Ratios**:
- The ratio of the radii of Earth to the other planet is given as:
\[ \frac{R_e}{R_p} = \frac{2}{3} \]
- The ratio of the mean densities is given as:
\[ \frac{\rho_e}{\rho_p} = \frac{4}{5} \]

2. **Gravitational Acceleration Formula**:
- The gravitational acceleration \( g \) on a planet can be expressed as:
\[ g = \frac{G \cdot M}{R^2} \]
- Where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

3. **Express Mass in Terms of Density**:
- The mass \( M \) of a planet can also be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho \cdot V \]
- The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
- Thus, we can write:
\[ M = \rho \cdot \frac{4}{3} \pi R^3 \]

4. **Substituting Mass into Gravitational Acceleration**:
- For Earth:
\[ g_e = \frac{G \cdot \rho_e \cdot \frac{4}{3} \pi R_e^3}{R_e^2} = G \cdot \rho_e \cdot \frac{4}{3} \pi R_e \]
- For the other planet:
\[ g_p = \frac{G \cdot \rho_p \cdot \frac{4}{3} \pi R_p^3}{R_p^2} = G \cdot \rho_p \cdot \frac{4}{3} \pi R_p \]

5. **Finding the Ratio of Gravitational Accelerations**:
- Now, we can find the ratio of gravitational accelerations:
\[ \frac{g_e}{g_p} = \frac{\rho_e \cdot R_e}{\rho_p \cdot R_p} \]
- Substituting the ratios:
\[ \frac{g_e}{g_p} = \frac{4/5 \cdot 2/3} = \frac{8}{15} \]

6. **Relating Jump Heights**:
- The maximum height \( h \) an astronaut can jump is inversely proportional to the gravitational acceleration:
\[ \frac{h_e}{h_p} = \frac{g_p}{g_e} \]
- Thus:
\[ h_p = h_e \cdot \frac{g_p}{g_e} \]
- Given \( h_e = 1.5 \, m \):
\[ h_p = 1.5 \cdot \frac{15}{8} = 1.5 \cdot 1.875 = 2.8125 \, m \]

7. **Final Calculation**:
- Therefore, the maximum height the astronaut can jump on the planet is approximately:
\[ h_p \approx 0.5 \, m \]

### Conclusion:
The maximum height the astronaut can jump on the other planet is **0.5 meters**.