A hammer of mass 1 kg strikes on Ihe head of a nail with a velocity of `2ms^(-1)` . It drives the nail 0.01 m into a wooden block. Find the force applied by the hammer and the time of impact.

Options

A. `200 N, 10^(-2)` sec
B. `100 N, 10^(-3)` sec
C. `300 N, 10^(-2)` sec
D. `400 N, 10^(-3)` sec

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given data
- Mass of the hammer (m) = 1 kg
- Initial velocity of the hammer (u) = 2 m/s
- Final velocity of the hammer (v) = 0 m/s (since the hammer comes to rest after driving the nail)
- Distance driven into the wooden block (s) = 0.01 m

### Step 2: Use the third equation of motion to find acceleration (a)
The third equation of motion is given by:
\[ v^2 = u^2 + 2as \]
Substituting the known values:
\[ 0^2 = (2)^2 + 2a(0.01) \]
\[ 0 = 4 + 0.02a \]
Rearranging the equation to solve for acceleration (a):
\[ 0.02a = -4 \]
\[ a = \frac{-4}{0.02} = -200 \, \text{m/s}^2 \]

### Step 3: Calculate the force applied by the hammer (F)
Using Newton's second law, the force can be calculated as:
\[ F = ma \]
Substituting the values:
\[ F = 1 \, \text{kg} \times (-200 \, \text{m/s}^2) \]
\[ F = -200 \, \text{N} \]
(The negative sign indicates that the force is in the opposite direction of the initial motion of the hammer.)

### Step 4: Calculate the time of impact (t)
Using the first equation of motion:
\[ v = u + at \]
Substituting the known values:
\[ 0 = 2 + (-200)t \]
Rearranging to solve for time (t):
\[ 200t = 2 \]
\[ t = \frac{2}{200} = 0.01 \, \text{s} \]

### Final Results
- Force applied by the hammer: \( F = 200 \, \text{N} \)
- Time of impact: \( t = 0.01 \, \text{s} \)

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