A metal plate of mass 200g is balanced in mid air by throwing 40 balls per second, each of mass 2g vertically upwards from below. The balls get rebounded with the same speed with which they strike the plate. Find the speed with which the balls strike the plate.

Answer

To solve the problem step by step, we will analyze the forces and momentum involved in the scenario.

### Step 1: Understand the Given Data
- Mass of the metal plate, \( M = 200 \, \text{g} = 0.2 \, \text{kg} \)
- Mass of each ball, \( m = 2 \, \text{g} = 0.002 \, \text{kg} \)
- Number of balls thrown per second, \( n = 40 \)

### Step 2: Calculate the Total Weight of the Plate
The weight of the plate can be calculated using the formula:
\[ W = M \cdot g \]
where \( g \) (acceleration due to gravity) is approximately \( 10 \, \text{m/s}^2 \).

Calculating the weight:
\[ W = 0.2 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 2 \, \text{N} \]

### Step 3: Determine the Change in Momentum of the Balls
When a ball strikes the plate with speed \( V \) and rebounds with the same speed \( V \), the change in momentum for one ball is:
\[ \Delta p = m \cdot V - (-m \cdot V) = 2mV \]
Since the ball rebounds with the same speed, the total change in momentum for \( n \) balls per second is:
\[ \Delta p_{\text{total}} = n \cdot \Delta p = n \cdot (2mV) = 40 \cdot (2 \cdot 0.002 \cdot V) = 0.16V \]

### Step 4: Set Up the Equation for Equilibrium
For the plate to be balanced in mid-air, the total upward momentum change (from the balls) must equal the weight of the plate:
\[ 0.16V = 2 \]

### Step 5: Solve for \( V \)
Now, we can solve for \( V \):
\[ V = \frac{2}{0.16} = 12.5 \, \text{m/s} \]

### Final Answer
The speed with which the balls strike the plate is \( V = 12.5 \, \text{m/s} \).

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