A particle of mass 15 kg has an initial velocity `vecv_(i) = hati - 2 hatjm//s` . It collides with another body and the impact time is 0.1s, resulting in a velocity `vecc_f = 6 hati + 4hatj + 5 hatk m//s` after impact. The average force of impact on the particle is :

Options

A. `15[5hati + 6hatj+ 5hatk]`
B. `15[5hati + 6hatj - 5hatk]`
C. `150[5hati - 6hatj+ 5hatk]`
D. `150[5hati + 6hatj+ 5hatk]`

Explanation

To find the average force of impact on the particle, we will follow these steps:

### Step 1: Calculate the initial momentum (P_i)
The initial momentum \( P_i \) can be calculated using the formula:
\[ P_i = m \cdot v_i \]
where:
- \( m = 15 \, \text{kg} \)
- \( v_i = \hat{i} - 2 \hat{j} \, \text{m/s} \)

Calculating \( P_i \):
\[ P_i = 15 \, \text{kg} \cdot (\hat{i} - 2 \hat{j}) = 15 \hat{i} - 30 \hat{j} \, \text{kg m/s} \]

### Step 2: Calculate the final momentum (P_f)
The final momentum \( P_f \) is given by:
\[ P_f = m \cdot v_f \]
where:
- \( v_f = 6 \hat{i} + 4 \hat{j} + 5 \hat{k} \, \text{m/s} \)

Calculating \( P_f \):
\[ P_f = 15 \, \text{kg} \cdot (6 \hat{i} + 4 \hat{j} + 5 \hat{k}) = 90 \hat{i} + 60 \hat{j} + 75 \hat{k} \, \text{kg m/s} \]

### Step 3: Calculate the change in momentum (ΔP)
The change in momentum \( \Delta P \) is given by:
\[ \Delta P = P_f - P_i \]
Calculating \( \Delta P \):
\[ \Delta P = (90 \hat{i} + 60 \hat{j} + 75 \hat{k}) - (15 \hat{i} - 30 \hat{j}) = (90 - 15) \hat{i} + (60 + 30) \hat{j} + 75 \hat{k} \]
\[ \Delta P = 75 \hat{i} + 90 \hat{j} + 75 \hat{k} \, \text{kg m/s} \]

### Step 4: Calculate the average force (F_avg)
The average force \( F_{avg} \) is given by:
\[ F_{avg} = \frac{\Delta P}{\Delta t} \]
where:
- \( \Delta t = 0.1 \, \text{s} \)

Calculating \( F_{avg} \):
\[ F_{avg} = \frac{75 \hat{i} + 90 \hat{j} + 75 \hat{k}}{0.1} = 750 \hat{i} + 900 \hat{j} + 750 \hat{k} \, \text{N} \]

### Final Answer
The average force of impact on the particle is:
\[ F_{avg} = 750 \hat{i} + 900 \hat{j} + 750 \hat{k} \, \text{N} \]

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