Question

A force $\vec{F} = 4\hat{i} - 15\hat{j}$ N acts on a body resulting in a displacement of $6\hat{i}$. If the body had kinetic energy of 7 joules at the beginning of the displacement, the kinetic energy at the end of the displacement is

• A. 24 J
• B. 31 J
• C. 30 J
• D. 25 J

Hint:

The work-energy theorem is a powerful tool to relate work done by forces to changes in kinetic energy. Always compute the dot product carefully when forces and displacements are given as vectors.

Answer 1

The Correct Option is B

To find the final kinetic energy, we use the work-energy theorem, which states that the work done by the net force equals the change in kinetic energy: \[ W = \Delta KE = KE_{\text{final}} - KE_{\text{initial}} \] 1. Calculate the work done:
- The force is $\vec{F} = 4\hat{i} - 15\hat{j}$ N, and the displacement is $\vec{d} = 6\hat{i}$ m.
- Work done $W = \vec{F} \cdot \vec{d}$ (dot product): \[ \begin{align} W = (4\hat{i} - 15\hat{j}) \cdot (6\hat{i}) = (4 \cdot 6) + (-15 \cdot 0) = 24 \, \text{J} \] 2. Apply the work-energy theorem:
- Initial kinetic energy $KE_{\text{initial}} = 7 \, \text{J}$.
- Work done $W = 24 \, \text{J}$.
- Using the work-energy theorem:
\[ \begin{align} W = KE_{\text{final}} - KE_{\text{initial}} \implies 24 = KE_{\text{final}} - 7 \implies KE_{\text{final}} = 24 + 7 = 31 \, \text{J} \] The final kinetic energy is 31 J, which matches option (2). Thus, the correct answer is (2).