Question

A force $ 6 \hat{k} $ is applied for $ \frac{5}{3} $ seconds on a body of mass 2 kg. If the initial velocity of the body was $ 3\hat{i} + 4\hat{j} $, then find the final velocity of the body.

• A. \( 3\hat{i} + \hat{j} + 5\hat{k} \)
• B. \( 3\hat{i} + 4\hat{j} + 5\hat{k} \)
• C. \( 3\hat{i} + 2\hat{j} - 3\hat{k} \)
• D. \( 3\hat{i} + 4\hat{j} - 5\hat{k} \)

Hint:

When a force is applied on an object, use Newton's second law to calculate the acceleration, and then apply the equation of motion to find the final velocity.

Answer 1

The Correct Option is B

Step 1: Apply Newton's Second Law.
Newton's second law states that the force \( \vec{F} \) on a body is equal to its mass \( m \) multiplied by its acceleration \( \vec{a} \):\[\vec{F} = m \vec{a}\]Given \( \vec{F} = 6 \hat{k} \, \text{N} \) and \( m = 2 \, \text{kg} \), the acceleration is calculated as:\[\vec{a} = \frac{\vec{F}}{m} = \frac{6 \hat{k}}{2} = 3 \hat{k} \, \text{m/s}^2\]
Step 2: Calculate final velocity.
The equation for final velocity \( \vec{v_f} \) is:\[\vec{v_f} = \vec{v_i} + \vec{a} \times t\]With the following values:\( \vec{v_i} = 3 \hat{i} + 4 \hat{j} \) (initial velocity),
\( \vec{a} = 3 \hat{k} \) (acceleration),
\( t = \frac{5}{3} \) seconds (time duration).
Substituting these values:\[\vec{v_f} = (3 \hat{i} + 4 \hat{j}) + (3 \hat{k}) \times \frac{5}{3}\]\[\vec{v_f} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k}\]
Step 3: Final Result.
The body's final velocity is \( 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \). This matches option (2).