How high a man be able to jump on the surface of a planet of radius 320 km, but having density same as that of the earth if he jumps 5 m on the surface of the earth? (Radius of earth = 6400 km)

Options

A. 60 m
B. 80 m
C. 100 m
D. 120 m

Answer

To solve the problem of how high a man can jump on a planet with a radius of 320 km and the same density as Earth, given that he can jump 5 m on Earth, we will follow these steps:

### Step 1: Understand the relationship between jump height and gravitational acceleration
The height a person can jump is directly related to the gravitational acceleration of the planet. The formula for the height of the jump is derived from the conservation of energy, where the kinetic energy at takeoff is converted to gravitational potential energy at the peak of the jump.

### Step 2: Calculate the gravitational acceleration on Earth
The gravitational acceleration on the surface of the Earth (g) can be calculated using the formula:
\[ g = \frac{GM}{R^2} \]
Where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

### Step 3: Calculate the gravitational acceleration on the new planet
Since the new planet has the same density as Earth, we can express its gravitational acceleration (g1) in terms of its radius (R1):
\[ g_1 = \frac{GM_1}{R_1^2} \]
Where \( M_1 \) can be expressed as:
\[ M_1 = \rho \cdot V_1 = \rho \cdot \left(\frac{4}{3} \pi R_1^3\right) \]
Thus, we can relate the gravitational accelerations:
\[ g_1 = \frac{G \cdot \rho \cdot \left(\frac{4}{3} \pi R_1^3\right)}{R_1^2} = \frac{4}{3} \pi G \rho R_1 \]

### Step 4: Relate the gravitational accelerations of Earth and the new planet
Since both planets have the same density, we can set up the ratio of gravitational accelerations:
\[ \frac{g_1}{g} = \frac{R_1}{R} \]
Where:
- \( R \) is the radius of Earth (6400 km),
- \( R_1 \) is the radius of the new planet (320 km).

### Step 5: Substitute the values
Substituting the values:
\[ \frac{g_1}{g} = \frac{320}{6400} = \frac{1}{20} \]
Thus, we find:
\[ g_1 = \frac{g}{20} \]

### Step 6: Calculate the height of the jump on the new planet
The height of the jump on the new planet (h1) can be calculated using the relationship:
\[ h_1 = \frac{g}{g_1} \cdot h \]
Where:
- \( h \) is the height jumped on Earth (5 m).

Substituting the values:
\[ h_1 = \frac{g}{\frac{g}{20}} \cdot 5 = 20 \cdot 5 = 100 \text{ m} \]

### Final Answer
The height a man can jump on the surface of the new planet is **100 meters**.

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