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. 1.24 m

Explanation

To solve the problem, we need to relate the maximum height an astronaut can jump on Earth to the maximum height he can jump on another planet using the given ratios of radii and densities.

### Step-by-Step Solution:

1. **Understanding the Energy Conversion**:
The kinetic energy (KE) given to the astronaut when he jumps is converted into gravitational potential energy (PE) at the maximum height. The relationship can be expressed as:
\[ KE = PE \]
\[ \frac{1}{2} mv^2 = mgh \]
where \( m \) is the mass of the astronaut, \( g \) is the acceleration due to gravity, and \( h \) is the height jumped.

2. **Setting Up the Equation for Earth**:
For Earth, the maximum height \( h_E \) the astronaut can jump is given as 1.5 m. Thus, we can write:
\[ m g_E h_E = \frac{1}{2} mv^2 \]
Simplifying, we get:
\[ g_E h_E = v^2 \]

3. **Setting Up the Equation for the Other Planet**:
For the other planet, let the maximum height be \( h_P \). The equation becomes:
\[ m g_P h_P = \frac{1}{2} mv^2 \]
Thus,
\[ g_P h_P = v^2 \]

4. **Equating the Two Heights**:
Since \( v^2 \) is the same in both cases, we can equate the two equations:
\[ g_E h_E = g_P h_P \]
Rearranging gives:
\[ h_P = \frac{g_E}{g_P} h_E \]

5. **Finding the Ratios**:
We need to find the ratio \( \frac{g_E}{g_P} \). The acceleration due to gravity can be expressed as:
\[ g = \frac{GM}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass, and \( R \) is the radius of the planet.

6. **Relating Mass to Density**:
The mass \( M \) can be expressed in terms of density \( \rho \) and volume \( V \):
\[ M = \rho V = \rho \left(\frac{4}{3} \pi R^3\right) \]
Thus,
\[ g = \frac{G \rho \left(\frac{4}{3} \pi R^3\right)}{R^2} = \frac{4}{3} \pi G \rho R \]

7. **Calculating the Ratio of Gravitational Accelerations**:
For Earth:
\[ g_E = \frac{4}{3} \pi G \rho_E R_E \]
For the other planet:
\[ g_P = \frac{4}{3} \pi G \rho_P R_P \]
The ratio becomes:
\[ \frac{g_E}{g_P} = \frac{\rho_E R_E}{\rho_P R_P} \]

8. **Substituting the Given Ratios**:
We know:
- The ratio of radii \( \frac{R_E}{R_P} = \frac{2}{3} \)
- The ratio of densities \( \frac{\rho_E}{\rho_P} = \frac{4}{5} \)

Therefore:
\[ \frac{g_E}{g_P} = \frac{4/5 \cdot 2/3} = \frac{8}{15} \]

9. **Calculating the Maximum Height on the Other Planet**:
Now substituting back into the height equation:
\[ h_P = \frac{g_E}{g_P} h_E = \frac{8}{15} \cdot 1.5 \]
\[ h_P = \frac{8 \cdot 1.5}{15} = \frac{12}{15} = 0.8 \text{ m} \]

### Final Answer:
The maximum height the astronaut can jump on the other planet is **0.8 meters**.