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 (with the same density as Earth) after jumping 5 m on Earth, we can follow these steps:

### Step 1: Understand the relationship between gravitational acceleration and radius
The gravitational acceleration \( g \) on the surface of a planet is given by the formula:
\[ g = \frac{G \cdot M}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

### Step 2: Relate the mass of the planet to its density
The mass \( M \) of the planet can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho \cdot V = \rho \cdot \frac{4}{3} \pi R^3 \]
Substituting this into the equation for \( g \):
\[ g = \frac{G \cdot \rho \cdot \frac{4}{3} \pi R^3}{R^2} = \frac{4}{3} \pi G \rho R \]
This shows that \( g \) is directly proportional to the radius \( R \) of the planet when density \( \rho \) is constant.

### Step 3: Calculate gravitational acceleration on Earth and the new planet
For Earth:
- Radius \( R_{earth} = 6400 \) km
- Gravitational acceleration \( g_{earth} \)

For the new planet:
- Radius \( R_{planet} = 320 \) km
- Gravitational acceleration \( g_{planet} \)

Using the proportionality:
\[ \frac{g_{planet}}{g_{earth}} = \frac{R_{planet}}{R_{earth}} = \frac{320}{6400} = \frac{1}{20} \]
Thus, \( g_{planet} = \frac{g_{earth}}{20} \).

### Step 4: Use the jump height relationship
The height a man can jump is related to the gravitational acceleration. If he jumps to a height \( h \) on Earth, we can relate the heights on both planets:
\[ h_{earth} = \frac{v^2}{2g_{earth}} \quad \text{and} \quad h_{planet} = \frac{v^2}{2g_{planet}} \]
From the above, we can express the height on the new planet:
\[ h_{planet} = h_{earth} \cdot \frac{g_{earth}}{g_{planet}} \]

### Step 5: Substitute known values
Given that \( h_{earth} = 5 \) m:
\[ h_{planet} = 5 \cdot \frac{g_{earth}}{g_{planet}} = 5 \cdot 20 = 100 \text{ m} \]

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

---