A body of mass 5 kg is moving with velocity of `v=(2hati+6hatj)ms^(-1)` at t=0s. After time t=2s, velocity of body is `(10 hati+6 hatj)`, then change in momentum to body is

Options

A. `40 hati kgms^(-1)`
B. `20hati kgms^(-1)`
C. `30 hati kgms^(-1)`
D. `(50 hati+30 hatj)kgms^(-1)`

Explanation

To find the change in momentum of the body, we will follow these steps:

### Step 1: Identify the given values
- Mass of the body (m) = 5 kg
- Initial velocity (v_i) at t = 0 s = \(2 \hat{i} + 6 \hat{j}\) m/s
- Final velocity (v_f) at t = 2 s = \(10 \hat{i} + 6 \hat{j}\) m/s

### Step 2: Calculate the initial momentum
The initial momentum (p_i) can be calculated using the formula:
\[ p_i = m \cdot v_i \]
Substituting the values:
\[ p_i = 5 \, \text{kg} \cdot (2 \hat{i} + 6 \hat{j}) \, \text{m/s} \]
\[ p_i = 5 \cdot 2 \hat{i} + 5 \cdot 6 \hat{j} \]
\[ p_i = 10 \hat{i} + 30 \hat{j} \, \text{kg m/s} \]

### Step 3: Calculate the final momentum
The final momentum (p_f) can be calculated using the formula:
\[ p_f = m \cdot v_f \]
Substituting the values:
\[ p_f = 5 \, \text{kg} \cdot (10 \hat{i} + 6 \hat{j}) \, \text{m/s} \]
\[ p_f = 5 \cdot 10 \hat{i} + 5 \cdot 6 \hat{j} \]
\[ p_f = 50 \hat{i} + 30 \hat{j} \, \text{kg m/s} \]

### Step 4: Calculate the change in momentum
The change in momentum (Δp) is given by:
\[ \Delta p = p_f - p_i \]
Substituting the values:
\[ \Delta p = (50 \hat{i} + 30 \hat{j}) - (10 \hat{i} + 30 \hat{j}) \]
\[ \Delta p = (50 - 10) \hat{i} + (30 - 30) \hat{j} \]
\[ \Delta p = 40 \hat{i} + 0 \hat{j} \, \text{kg m/s} \]

### Step 5: Final answer
The change in momentum of the body is:
\[ \Delta p = 40 \hat{i} \, \text{kg m/s} \]

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