A hammer weighing 3 kg, moving with a velocity of 10m/s, strikes against the hed of a spike and drives it into a block of wood. If the hammer comes to rest in 0.025 s, the impulse associated with the ball will be

Options

A. 30Ns
B. `-30Ns`
C. `15Ns`
D. `-15Ns`

Answer

To solve the problem of finding the impulse associated with the hammer, we will follow these steps:

### Step 1: Understand the concept of impulse
Impulse is defined as the change in momentum of an object. It can be mathematically expressed as:
\[ \text{Impulse} (I) = \Delta p = p_{\text{final}} - p_{\text{initial}} \]
where \( p \) is the momentum.

### Step 2: Calculate the initial momentum
The initial momentum of the hammer can be calculated using the formula:
\[ p_{\text{initial}} = m \times v \]
where:
- \( m = 3 \, \text{kg} \) (mass of the hammer)
- \( v = 10 \, \text{m/s} \) (initial velocity of the hammer)

Calculating this gives:
\[ p_{\text{initial}} = 3 \, \text{kg} \times 10 \, \text{m/s} = 30 \, \text{kg m/s} \]

### Step 3: Calculate the final momentum
When the hammer comes to rest, its final velocity is 0 m/s. Therefore, the final momentum is:
\[ p_{\text{final}} = m \times v_{\text{final}} = 3 \, \text{kg} \times 0 \, \text{m/s} = 0 \, \text{kg m/s} \]

### Step 4: Calculate the change in momentum
Now, we can calculate the change in momentum:
\[ \Delta p = p_{\text{final}} - p_{\text{initial}} \]
Substituting the values we found:
\[ \Delta p = 0 \, \text{kg m/s} - 30 \, \text{kg m/s} = -30 \, \text{kg m/s} \]

### Step 5: Determine the impulse associated with the hammer
Since the impulse is equal to the change in momentum, we have:
\[ I = \Delta p = -30 \, \text{kg m/s} \]

### Conclusion
The impulse associated with the hammer (and thus with the spike) is:
\[ \text{Impulse} = -30 \, \text{N s} \]

### Final Answer
The impulse associated with the spike is \(-30 \, \text{N s}\).

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