A juggler keeps n balls going with one hand, so that at any instant, (n - 1) balls are in air and one ball in the hand. If each ball rises to a height of x metres, the time for each ball to stay in his hand is

Options

A. `1/n -1((sqrt 2x)/g)`
B. `2/n - 1((sqrt 2x)/g)`
C. `2/n((sqrt 2x)/ g)`
D. `1/n((sqrt 2x)/ g)`

Explanation

To solve the problem, we need to determine the time each ball stays in the juggler's hand when juggling \( n \) balls, where each ball rises to a height of \( x \) meters. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Motion of the Ball**:
- When a ball is thrown upwards, it rises to a maximum height \( x \) before coming back down. At the maximum height, the final velocity \( v = 0 \).

2. **Using the Kinematic Equation**:
- We can use the kinematic equation:
\[ v^2 = u^2 + 2as \]
where:
- \( v = 0 \) (final velocity at the height),
- \( u \) is the initial velocity,
- \( a = -g \) (acceleration due to gravity, acting downwards),
- \( s = x \) (the height reached).

- Plugging in the values, we get:
\[ 0 = u^2 - 2gx \]
This simplifies to:
\[ u^2 = 2gx \quad \Rightarrow \quad u = \sqrt{2gx} \]

3. **Calculating Time to Reach Maximum Height**:
- The time \( t \) taken to reach the maximum height can be found using the equation:
\[ v = u + at \]
Substituting the known values:
\[ 0 = u - gt \quad \Rightarrow \quad gt = u \quad \Rightarrow \quad t = \frac{u}{g} \]
- Substituting \( u = \sqrt{2gx} \):
\[ t = \frac{\sqrt{2gx}}{g} = \frac{\sqrt{2x}}{\sqrt{g}} \]

4. **Total Time for the Ball to Go Up and Come Down**:
- The total time \( T \) for the ball to go up and come back down is:
\[ T = 2t = 2 \cdot \frac{\sqrt{2x}}{\sqrt{g}} = \frac{2\sqrt{2x}}{\sqrt{g}} \]

5. **Time Each Ball Stays in the Juggler's Hand**:
- Since there are \( n \) balls and at any instant, \( n - 1 \) balls are in the air, the time each ball stays in the juggler's hand is given by:
\[ \text{Time in hand} = \frac{T}{n - 1} = \frac{2\sqrt{2x}}{\sqrt{g}(n - 1)} \]

### Final Answer:
The time for each ball to stay in the juggler's hand is:
\[ \text{Time in hand} = \frac{2\sqrt{2x}}{\sqrt{g}(n - 1)} \]