Question

an object of mass ( m ) is moving to the right along a horizontal surface with speed ( v_{i} ) when a force of magnitude ( f ) is exerted on it, where ( f<mg ). in which of the following situations is the rate of energy being transferred to the object the greatest?

Explanation:

Step1: Recall the formula for power

Power \( P=\vec{F}\cdot\vec{v}=Fv\cos\theta\), where \(\theta\) is the angle between the force \(\vec{F}\) and the velocity \(\vec{v}\).

Step2: Analyze each option

• Option A: The force is vertical (\(\theta = 90^{\circ}\)), so \(P = Fv_{i}\cos90^{\circ}=0\) (since \(\cos90^{\circ}=0\)).
• Option B: The force is opposite to the velocity (\(\theta = 180^{\circ}\)), so \(P = Fv_{i}\cos180^{\circ}=-Fv_{i}\) (negative sign indicates energy is being transferred away from the object).
• Option C: The force is in the same direction as the velocity (\(\theta = 0^{\circ}\)), so \(P = Fv_{i}\cos0^{\circ}=Fv_{i}\) (since \(\cos0^{\circ} = 1\)).
• Option D: The force has a vertical component. Let the angle between \(F\) and the horizontal be \(\alpha\) (\(0<\alpha<90^{\circ}\)). Then \(P = Fv_{i}\cos\alpha\), and since \(\cos\alpha<1\) for \(0<\alpha<90^{\circ}\), \(P

Answer:

C.