A body of mass 2kg is moving along positive X - direction with a velocity of `5 ms^(-1)` Now a force of `10sqrt(2)` N N is applied at an angle 45° with X - axis. Its velocity after 3s is,

Options

A. `20 ms^(-1)`
B. `15 ms^(-1)`
C. `25 ms^(-1)`
D. `5 ms^(-1)`

Answer

To solve the problem step by step, we will follow the principles of Newton's laws of motion and vector addition.

### Step 1: Identify the given values
- Mass of the body, \( m = 2 \, \text{kg} \)
- Initial velocity in the x-direction, \( u_x = 5 \, \text{m/s} \)
- Force applied, \( F = 10\sqrt{2} \, \text{N} \)
- Angle of force with the x-axis, \( \theta = 45^\circ \)
- Time duration, \( t = 3 \, \text{s} \)

### Step 2: Calculate the acceleration
To find the acceleration, we first need to resolve the force into its components. The force in the x-direction and y-direction can be calculated as follows:

- \( F_x = F \cdot \cos(\theta) = 10\sqrt{2} \cdot \cos(45^\circ) = 10\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 10 \, \text{N} \)
- \( F_y = F \cdot \sin(\theta) = 10\sqrt{2} \cdot \sin(45^\circ) = 10\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 10 \, \text{N} \)

Now, we can find the acceleration in both directions using Newton's second law \( F = ma \):

- \( a_x = \frac{F_x}{m} = \frac{10}{2} = 5 \, \text{m/s}^2 \)
- \( a_y = \frac{F_y}{m} = \frac{10}{2} = 5 \, \text{m/s}^2 \)

### Step 3: Calculate the final velocity in the x-direction
Using the equation of motion:
\[ v_x = u_x + a_x t \]
Substituting the known values:
\[ v_x = 5 + 5 \cdot 3 = 5 + 15 = 20 \, \text{m/s} \]

### Step 4: Calculate the final velocity in the y-direction
Since the initial velocity in the y-direction is zero:
\[ v_y = u_y + a_y t \]
Substituting the known values:
\[ v_y = 0 + 5 \cdot 3 = 0 + 15 = 15 \, \text{m/s} \]

### Step 5: Calculate the resultant velocity
The resultant velocity can be found using the Pythagorean theorem:
\[ v = \sqrt{v_x^2 + v_y^2} \]
Substituting the values:
\[ v = \sqrt{20^2 + 15^2} = \sqrt{400 + 225} = \sqrt{625} = 25 \, \text{m/s} \]

### Final Answer
The velocity of the body after 3 seconds is \( 25 \, \text{m/s} \).

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