Question

An object moving along horizontal x-direction with kinetic energy \(10 \, \text{J}\) is displaced through \(\mathbf{x} = (3\mathbf{i}) \, \text{m}\) by the force \(\mathbf{F} = (-2\mathbf{i} + 3\mathbf{j}) \, \text{N}\). The kinetic energy of the object at the end of the displacement \(\mathbf{x}\) is:

• A. \(10 \, \text{J}\)
• B. \(16 \, \text{J}\)
• C. \(4 \, \text{J}\)
• D. \(6 \, \text{J}\)

Answer 1

The Correct Option is C

To determine the kinetic energy of the object at the end of the displacement, we need to consider the work done by the force on the object and how it affects the kinetic energy. 

The work-energy principle states that the work done by all forces acting on an object is equal to the change in its kinetic energy. Mathematically, this is represented as:

\(W = \Delta KE = KE_{\text{final}} - KE_{\text{initial}}\)

where \( W \) is the work done by the force, \(\Delta KE\) is the change in kinetic energy, \( KE_{\text{final}} \) is the final kinetic energy, and \( KE_{\text{initial}} \) is the initial kinetic energy.

Given:

• Initial kinetic energy, \( KE_{\text{initial}} = 10 \, \text{J} \)
• Displacement, \(\mathbf{x} = 3\mathbf{i} \, \text{m} \)
• Force, \(\mathbf{F} = (-2\mathbf{i} + 3\mathbf{j}) \, \text{N} \)

The work done by the force, \( W \), is calculated using the dot product of the force vector and the displacement vector:

\(W = \mathbf{F} \cdot \mathbf{x} = (-2\mathbf{i} + 3\mathbf{j}) \cdot (3\mathbf{i})\)

Simplifying the dot product:

\(W = (-2)(3) + (3)(0) = -6 \, \text{J}\)

This indicates that the work done by the force is \(-6 \, \text{J}\). The negative sign shows that the force reduces the kinetic energy of the object.

Substituting the values in the work-energy principle:

\(-6 \, \text{J} = KE_{\text{final}} - 10 \, \text{J}\)

Simplifying, we find:

\(KE_{\text{final}} = -6 + 10 = 4 \, \text{J}\)

Therefore, the kinetic energy of the object at the end of the displacement is \(4 \, \text{J}\).

The correct answer is: \(4 \, \text{J}\).