A force `2hati+hatj-hatk` newton acts on a body which is initially at rest.If the velocity of the body at the end of `20seconds` is `4hati+2hatj+2hatk ms^(-1)`, the mass of the body

Options

A. `20kg`
B. `15kg`
C. `10kg`
D. `5kg`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the Given Information
We have a force acting on a body given by:
\[ \vec{F} = 2\hat{i} + \hat{j} - \hat{k} \text{ N} \]
The initial velocity of the body is:
\[ \vec{u} = 0 \text{ m/s} \quad (\text{since the body is initially at rest}) \]
The final velocity after 20 seconds is:
\[ \vec{v} = 4\hat{i} + 2\hat{j} + 2\hat{k} \text{ m/s} \]
The time duration is:
\[ t = 20 \text{ s} \]

### Step 2: Calculate the Acceleration
Using Newton's second law, the acceleration \(\vec{a}\) can be found using the formula:
\[ \vec{F} = m \vec{a} \]
Thus, the acceleration can be expressed as:
\[ \vec{a} = \frac{\vec{F}}{m} \]

### Step 3: Use the Equation of Motion
The equation of motion relating initial velocity, final velocity, acceleration, and time is:
\[ \vec{v} = \vec{u} + \vec{a} t \]
Substituting \(\vec{u} = 0\):
\[ \vec{v} = \vec{a} t \]
This implies:
\[ \vec{a} = \frac{\vec{v}}{t} \]

### Step 4: Calculate the Magnitude of the Final Velocity
The magnitude of the final velocity \(\vec{v}\) is:
\[ |\vec{v}| = \sqrt{(4)^2 + (2)^2 + (2)^2} = \sqrt{16 + 4 + 4} = \sqrt{24} = 2\sqrt{6} \text{ m/s} \]

### Step 5: Calculate the Magnitude of the Force
The magnitude of the force \(\vec{F}\) is:
\[ |\vec{F}| = \sqrt{(2)^2 + (1)^2 + (-1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \text{ N} \]

### Step 6: Relate Force, Mass, and Acceleration
From the relation between force, mass, and acceleration:
\[ \vec{a} = \frac{\vec{F}}{m} \]
We can express \(m\) as:
\[ m = \frac{|\vec{F}| \cdot t}{|\vec{v}|} \]

### Step 7: Substitute the Values
Substituting the values we calculated:
\[ m = \frac{|\vec{F}| \cdot t}{|\vec{v}|} = \frac{\sqrt{6} \cdot 20}{2\sqrt{6}} = \frac{20}{2} = 10 \text{ kg} \]

### Final Answer
The mass of the body is:
\[ \boxed{10 \text{ kg}} \]

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