Hammer of mass M strikes a nail of mass 'm' with a velocity 20 m/s into a fixed wall. The nail penetrates into the wall to a depth of 1 cm. The average resistance of the wall to the penetration of the nail is

Options

A. (a)`(M^2/(m+m))xx10^3`
B. (b)`(2M^2)/(M+m)xx10^4`
C. (c)`(M+m)/(M^2)xx10^2`
D. (d)`(M^2)/(M+m)xx10^2`

Answer

To find the average resistance of the wall to the penetration of the nail, we can follow these steps:

### Step 1: Understand the Problem
We have a hammer of mass \( M \) striking a nail of mass \( m \) with an initial velocity of \( 20 \, \text{m/s} \). The nail penetrates a wall to a depth of \( 1 \, \text{cm} \) (which is \( 0.01 \, \text{m} \)). We need to find the average resistance force exerted by the wall on the nail.

### Step 2: Use Conservation of Momentum
When the hammer strikes the nail, it transfers its momentum to the nail. Since the collision is inelastic (the hammer and nail stick together), we can use the conservation of momentum:

\[ M \cdot 20 + m \cdot 0 = (M + m) \cdot V \]

Where \( V \) is the final velocity after the collision.

### Step 3: Solve for Final Velocity \( V \)
From the momentum equation, we can solve for \( V \):

\[ V = \frac{M \cdot 20}{M + m} \]

### Step 4: Calculate the Deceleration
Next, we need to find the deceleration of the nail as it penetrates the wall. We can use the kinematic equation:

\[ V^2 = u^2 + 2a s \]

Where:
- \( V \) is the final velocity (0 m/s, as the nail comes to rest),
- \( u \) is the initial velocity (which we found as \( \frac{M \cdot 20}{M + m} \)),
- \( a \) is the acceleration (deceleration in this case),
- \( s \) is the distance penetrated (0.01 m).

Rearranging gives:

\[ 0 = \left(\frac{M \cdot 20}{M + m}\right)^2 + 2a(0.01) \]

### Step 5: Solve for Acceleration \( a \)
Rearranging the equation to solve for \( a \):

\[ a = -\frac{\left(\frac{M \cdot 20}{M + m}\right)^2}{2 \cdot 0.01} \]

### Step 6: Calculate the Average Resistance Force
The average resistance force \( F \) exerted by the wall can be calculated using Newton's second law:

\[ F = (M + m) \cdot a \]

Substituting \( a \):

\[ F = (M + m) \cdot \left(-\frac{\left(\frac{M \cdot 20}{M + m}\right)^2}{2 \cdot 0.01}\right) \]

### Step 7: Simplify the Expression
Now we simplify the expression for \( F \):

\[ F = -\frac{(M + m) \cdot (M^2 \cdot 400)}{(M + m)^2 \cdot 2 \cdot 0.01} \]

This simplifies to:

\[ F = -\frac{400M^2}{2 \cdot 0.01 \cdot (M + m)} \]

\[ F = -\frac{20000M^2}{(M + m)} \]

### Final Result
The average resistance of the wall to the penetration of the nail is:

\[ F = \frac{20000M^2}{(M + m)} \, \text{N} \]