Question

If a body at rest undergoes displacement under the action of force with constant acceleration, then the power delivered by the force at any time t is proportional to

• A. \(t\)
• B. \(\sqrt{t}\)
• C. \(t^2\)
• D. \(1 / t\)
• E. \(t^3\)

Hint:

For constant acceleration:
Velocity \(v \propto t\).
Displacement \(S \propto t^2\).
Power \(P \propto t\).
Work \(W \propto t^2\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Power (\(P\)) is the rate of doing work and is given by the product of force (\(F\)) and instantaneous velocity (\(v\)).

Step 2: Key Formula or Approach:
1. Newton's second law: \(F = ma\).
2. Equation of motion: \(v = u + at\).
3. Power: \(P = F \cdot v\).

Step 3: Detailed Explanation:
Given the body starts from rest, initial velocity \(u = 0\).
Under constant acceleration \(a\), the velocity at time \(t\) is:
\[ v = 0 + at = at \]
The force applied is constant since acceleration is constant:
\[ F = ma \]
The instantaneous power \(P\) delivered is:
\[ P = F \times v = (ma) \times (at) = ma^2 t \]
Since \(m\) and \(a\) are constant, we find:
\[ P \propto t \]

Step 4: Final Answer:
Power is proportional to \(t\).