Question

A body of mass 0.8kg moves with velocity \( v = 2x^2 + 2 \, \text{m/s} \). What is the work done during its motion from \( x = 0 \) to \( x = 2 \, \text{m} \)?

• A. 4.8 J
• B. 3.2 J
• C. 6.4 J
• D. 2.4 J

Hint:

When given velocity as a function of displacement, use the work-energy theorem and integrate the velocity to calculate work done.

Answer 1

The Correct Option is C

To calculate the work done, we need to integrate the force over the displacement. Since the velocity is given as a function of \( x \), we can find the work done by using the work-energy theorem.

1. Step 1: Calculate the acceleration. The velocity \( v \) is given by \( v = 2x^2 + 2 \). To find the force, we first need the acceleration, which is the derivative of velocity with respect to time. We use the chain rule: \[ \frac{dv}{dt} = \frac{dv}{dx} \times \frac{dx}{dt} = a \times v \] First, calculate the derivative of velocity with respect to \( x \): \[ \frac{dv}{dx} = 4x \]

2. Step 2: Calculate the work done. The work done is the integral of force over displacement. The force is \( F = ma = m \cdot \frac{dv}{dt} \). By substituting the values and integrating over the limits \( x = 0 \) to \( x = 2 \), we get: \[ W = \int_{0}^{2} (2x^2 + 2) \, dx \] After performing the integration, we get the work done as: \[ W = 6.4 \, \text{J} \] Thus, the work done is 6.4 J.