A body has an initial velocity of `3m//s` and has an acceleration of `1m//sec^(2)` normal to the direction of the initial velocity. Then its velocity 4 seconds after the start is

Options

A. `7m//sec` along the direction of initial velocity
B. `7m//sec` along the normal to the direction of initial velocity
C. `7m//sec` mid-way between the two directions
D. `5m//sec` at an angle of `tan^(-1)(4//3)` with the direction of initial velocity.

Explanation

To solve the problem, we need to determine the final velocity of a body that has an initial velocity of \(3 \, \text{m/s}\) and an acceleration of \(1 \, \text{m/s}^2\) acting normal (perpendicular) to the direction of the initial velocity after \(4\) seconds.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Initial velocity (\(u\)) in the x-direction: \(u_x = 3 \, \text{m/s}\)
- Initial velocity in the y-direction: \(u_y = 0 \, \text{m/s}\)
- Acceleration (\(a\)) in the y-direction: \(a_y = 1 \, \text{m/s}^2\)
- Time (\(t\)): \(4 \, \text{s}\)

2. **Determine the Final Velocity in the x-direction:**
- Since there is no acceleration in the x-direction, the velocity in the x-direction remains constant.
\[ v_x = u_x = 3 \, \text{m/s} \]

3. **Determine the Final Velocity in the y-direction:**
- Use the equation of motion for the y-direction:
\[ v_y = u_y + a_y \cdot t \]
- Substituting the values:
\[ v_y = 0 + (1 \, \text{m/s}^2) \cdot (4 \, \text{s}) = 4 \, \text{m/s} \]

4. **Calculate the Magnitude of the Final Velocity:**
- The final velocity (\(v\)) can be calculated using the Pythagorean theorem since the x and y components are perpendicular:
\[ v = \sqrt{v_x^2 + v_y^2} \]
- Substituting the values:
\[ v = \sqrt{(3 \, \text{m/s})^2 + (4 \, \text{m/s})^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{m/s} \]

5. **Determine the Angle of the Final Velocity:**
- The angle (\(\theta\)) with respect to the x-axis can be found using:
\[ \tan(\theta) = \frac{v_y}{v_x} = \frac{4}{3} \]
- Therefore, the angle is:
\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]

### Final Result:
- The magnitude of the final velocity after \(4\) seconds is \(5 \, \text{m/s}\).
- The angle with respect to the x-axis is \(\tan^{-1}\left(\frac{4}{3}\right)\).