A bullet of mass `m`m penetrates a thickness `h` of a fixed plate of mass `M`. If the plate was free to move, then the thickness penetrated will be

Options

A. (a) `(Mh)/(M+m)`
B. (b) `(2Mh)/(M+m)`
C. (c) `(mh)/(2(M+m))`
D. (d) `(Mh)/(2(M+m))`

Explanation

To solve the problem of how far a bullet penetrates a plate when the plate is free to move, we will analyze the situation step by step.

### Step 1: Understand the scenario
When a bullet of mass `m` strikes a fixed plate of mass `M`, it penetrates a thickness `h`. We need to determine how much the bullet will penetrate if the plate is free to move.

### Step 2: Analyze the fixed plate scenario
In the first scenario, where the plate is fixed:
- The bullet comes to a stop after penetrating the plate.
- The average retarding force acting on the bullet can be defined as \( F_{\text{avg}} \).
- Using the equation of motion, we can write:
\[ v^2 = u^2 + 2a s \]
where \( v = 0 \) (final velocity), \( u \) is the initial velocity of the bullet, \( a \) is the acceleration (which will be negative), and \( s = h \) (the penetration depth).

### Step 3: Calculate the acceleration
From the above equation, we can rearrange it to find the acceleration:
\[ 0 = u^2 + 2(-\frac{F_{\text{avg}}}{m})h \]
This simplifies to:
\[ u^2 = \frac{2F_{\text{avg}}h}{m} \]
This is our **Equation 1**.

### Step 4: Analyze the moving plate scenario
Now, consider the second scenario where the plate is free to move:
- When the bullet strikes the plate, both the bullet and the plate will move together after the impact.
- Let \( v \) be the final velocity of both the bullet and the plate after the collision.

### Step 5: Apply conservation of momentum
Using the principle of conservation of momentum:
\[ mu = (m + M)v \]
From this, we can express the final velocity \( v \):
\[ v = \frac{mu}{m + M} \]

### Step 6: Calculate the relative acceleration
The forces acting on the bullet and the plate will create relative acceleration. The acceleration of the bullet while penetrating the plate is:
\[ a_{\text{bullet}} = -\frac{F_{\text{avg}}}{m} \]
The acceleration of the plate is:
\[ a_{\text{plate}} = \frac{F_{\text{avg}}}{M} \]
The relative acceleration \( a_r \) between the bullet and the plate is:
\[ a_r = a_{\text{bullet}} - a_{\text{plate}} = -\frac{F_{\text{avg}}}{m} - \frac{F_{\text{avg}}}{M} \]

### Step 7: Calculate the penetration depth
Using the relative motion equation:
\[ v_r^2 = u_r^2 + 2a_r s_r \]
Where \( v_r = 0 \) when both the bullet and plate move together, and \( u_r = u \). Thus:
\[ 0 = u^2 + 2a_r s_r \]
Substituting for \( a_r \):
\[ 0 = u^2 + 2\left(-\frac{F_{\text{avg}}}{m} - \frac{F_{\text{avg}}}{M}\right)s_r \]
This leads to:
\[ s_r = \frac{mu^2}{2F_{\text{avg}}(1 + \frac{m}{M})} \]

### Step 8: Final expression for penetration depth
The penetration depth when the plate is free to move can be expressed as:
\[ s_r = \frac{h}{1 + \frac{m}{M}} \]

### Conclusion
Thus, the thickness penetrated by the bullet when the plate is free to move is given by:
\[ \text{Thickness penetrated} = \frac{h}{1 + \frac{m}{M}} \]