A planet has mass `1//10` of that of earth, while radius is `1//3` that of earth. If a person can throw a stone on earth surface height of 90m, then he will be able to throw the stone on that planet to a height

Options

A. `90m`
B. `40m`
C. `100m`
D. `45m`

Explanation

To solve the problem, we need to determine how high a person can throw a stone on a planet with a mass of \( \frac{1}{10} \) that of Earth and a radius of \( \frac{1}{3} \) that of Earth. We know that the person can throw a stone to a height of 90 meters on Earth.

### Step-by-Step Solution:

1. **Identify the formula for maximum height**:
The maximum height \( H \) to which a stone can be thrown is given by the formula:
\[ H = \frac{U^2}{2g} \]
where \( U \) is the initial velocity of the throw and \( g \) is the acceleration due to gravity.

2. **Define variables for Earth and the planet**:
Let:
- \( H_e = 90 \, \text{m} \) (height on Earth)
- \( g_e \) = acceleration due to gravity on Earth
- \( H_p \) = height on the planet
- \( g_p \) = acceleration due to gravity on the planet

3. **Set up the ratio of heights**:
Using the formula for maximum height, we can set up the ratio:
\[ \frac{H_e}{H_p} = \frac{U^2 / (2g_e)}{U^2 / (2g_p)} \]
This simplifies to:
\[ \frac{H_e}{H_p} = \frac{g_p}{g_e} \]

4. **Rearranging the equation**:
Rearranging gives us:
\[ H_p = H_e \cdot \frac{g_e}{g_p} \]

5. **Calculate the ratio of gravitational accelerations**:
The acceleration due to gravity \( g \) is given by:
\[ g = \frac{G \cdot M}{R^2} \]
Thus, the ratio of gravitational accelerations on Earth and the planet is:
\[ \frac{g_e}{g_p} = \frac{M_e / R_e^2}{M_p / R_p^2} \]
Given:
- \( M_p = \frac{1}{10} M_e \)
- \( R_p = \frac{1}{3} R_e \)

Substituting these values into the ratio:
\[ \frac{g_e}{g_p} = \frac{M_e / R_e^2}{(1/10) M_e / (1/3)^2 R_e^2} = \frac{1}{(1/10)} \cdot \frac{(1/3)^2}{1} = 10 \cdot \frac{1}{9} = \frac{10}{9} \]

6. **Substituting back to find \( H_p \)**:
Now substituting back into the equation for \( H_p \):
\[ H_p = 90 \cdot \frac{10}{9} = 100 \, \text{m} \]

### Final Answer:
The height to which the person can throw the stone on the planet is **100 meters**.