The acceleration due to gravity on the planet `A` is `9` times the acceleration due to gravity on planet `B`. A man jumps to a height of `2m` on the surface of `A`. What is the height of jump by the same person on the planet `B`?

Options

A. `2/3m`
B. `2/9m`
C. `18m`
D. `6m`

Explanation

To solve the problem, we will use the principle of conservation of energy. The kinetic energy when the man jumps will be converted into potential energy at the peak of his jump.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The acceleration due to gravity on planet A (\(g_A\)) is 9 times that on planet B (\(g_B\)).
- The height of the jump on planet A is \(h_A = 2 \, \text{m}\).

2. **Write the Energy Conservation Equation for Planet A:**
- The kinetic energy (KE) at the time of the jump is equal to the potential energy (PE) at the maximum height.
- The equation can be written as:
\[ \frac{1}{2} m v^2 = m g_A h_A \]
- Here, \(m\) is the mass of the man, \(v\) is the velocity at the time of the jump, and \(h_A\) is the height of the jump on planet A.

3. **Substituting Known Values for Planet A:**
- We know \(h_A = 2 \, \text{m}\) and \(g_A = 9 g_B\).
- Substituting these values into the equation gives:
\[ \frac{1}{2} m v^2 = m (9 g_B) (2) \]
- Simplifying this, we get:
\[ \frac{1}{2} v^2 = 18 g_B \]

4. **Write the Energy Conservation Equation for Planet B:**
- For planet B, the kinetic energy will also be equal to the potential energy at the maximum height \(h_B\):
\[ \frac{1}{2} m v^2 = m g_B h_B \]

5. **Equating the Kinetic Energies:**
- Since the kinetic energy at the time of the jump is the same on both planets, we can set the two equations equal to each other:
\[ 18 g_B = g_B h_B \]

6. **Solving for \(h_B\):**
- Dividing both sides by \(g_B\) (assuming \(g_B \neq 0\)):
\[ h_B = 18 \, \text{m} \]

Thus, the height of the jump by the same person on planet B is **18 meters**.