Question

A body of mass 8 kg is moved by a force \(F = 3x\) N, where x is the distance covered. Initial position is x = 2 m and the final position is x = 10 m. The initial speed is zero. The final speed is

• A. 6 m/s
• B. 12 m/s
• C. 18 m/s
• D. 14 m/s

Hint:

For variable force, integrate \(F\,dx = m v\,dv\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Work done by variable force = change in kinetic energy.
Step 2: Detailed Explanation:
\(F = 8v \frac{dv}{dx} \rightarrow 3x = 8v \frac{dv}{dx}\)
\(3x\,dx = 8v\,dv\)
Integrate: \(3\left[\frac{x^2}{2}\right]_{2}^{10} = 8\left[\frac{v^2}{2}\right]_{0}^{v}\)
\(\frac{3}{2}(100 - 4) = 4v^2\)
\(\frac{3}{2} \times 96 = 4v^2\)
\(144 = 4v^2 \rightarrow v = 6\ \mathrm{m/s}\)
Step 3: Final Answer:
Final speed is 6 m/s.