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:
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- \( 3 \hat{i} + 4 \hat{j} - 5 \hat{k} \)
- \( 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \)
- \( 3 \hat{i} + 4 \hat{j} + \sqrt{5} \hat{k} \)
- \( 4 \hat{i} + 3 \hat{j} + 5 \hat{k} \)
The Correct Option is B
Approach Solution - 1
To solve this question, we need to find the velocity of a body after it has been influenced by a constant force for a specific duration.
Given:
- Initial velocity of the body: \(\vec{v}_{\text{in}} = 3 \hat{i} + 4 \hat{j} \, \text{ms}^{-1}\)
- Mass of the body: \(m = 2 \, \text{kg}\)
- Force acting on the body: \(\vec{F} = 6 \hat{k} \, \text{N}\)
- Time duration in the force field: \(t = \frac{5}{3} \, \text{s}\)
To find the final velocity, we can use the equation of motion under constant acceleration:
\(\vec{v}_{\text{final}} = \vec{v}_{\text{in}} + \vec{a} \cdot t\)
First, let's calculate the acceleration using Newton's second law:
\(\vec{F} = m \cdot \vec{a} \Longrightarrow \vec{a} = \frac{\vec{F}}{m} = \frac{6 \hat{k}}{2} = 3 \hat{k} \, \text{ms}^{-2}\)
Substitute the values into the velocity equation:
\(\vec{v}_{\text{final}} = (3 \hat{i} + 4 \hat{j}) + (3 \hat{k}) \cdot \frac{5}{3}\)
Further simplify the equation:
\(\vec{v}_{\text{final}} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k}\)
Thus, the velocity of the body when it emerges from the force field is: \(3 \hat{i} + 4 \hat{j} + 5 \hat{k}\).
Hence, the correct answer is:
\( 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \)
Approach Solution -2
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