Question

A body of mass 2 kg moving with velocity of $ \vec{v}_{\text{in}} = 3 \hat{i} + 4 \hat{j} \, \text{ms}^{-1} $ enters into a constant force field of 6N directed along positive z-axis. If the body remains in the field for a period of $ \frac{5}{3} $ seconds, then velocity of the body when it emerges from force field is:

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

Hint:

In problems involving constant forces, use Newton's second law to find acceleration and apply the equations of motion to find the final velocity.

Answer 1

The Correct Option is B

Determine the body's final velocity after experiencing a constant force for a given time.

Given:

• Initial velocity: \(\vec{v}_{\text{in}} = 3 \hat{i} + 4 \hat{j} \, \text{ms}^{-1}\)
• Mass: \(m = 2 \, \text{kg}\)
• Force: \(\vec{F} = 6 \hat{k} \, \text{N}\)
• Time: \(t = \frac{5}{3} \, \text{s}\)

The final velocity is calculated using the equation of motion:

\(\vec{v}_{\text{final}} = \vec{v}_{\text{in}} + \vec{a} \cdot t\)

First, calculate acceleration using Newton's second law:

\(\vec{a} = \frac{\vec{F}}{m} = \frac{6 \hat{k}}{2} = 3 \hat{k} \, \text{ms}^{-2}\)

Substitute values into the velocity equation:

\(\vec{v}_{\text{final}} = (3 \hat{i} + 4 \hat{j}) + (3 \hat{k}) \cdot \frac{5}{3}\)

Simplify:

\(\vec{v}_{\text{final}} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k}\)

The final velocity is: \(3 \hat{i} + 4 \hat{j} + 5 \hat{k}\).

The correct answer is:

\( 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \)