Question

A motor drives a body along a straight line with a constant force. The power \(P\) developed by the motor must vary with time \(t\) as:

• A. Option a
• B. Option b
• C. Option c
• D. Option d

Hint:

Key relations to remember:
Constant force \(\Rightarrow\) constant acceleration
\(v \propto t\) (if starting from rest)
Power \(P = Fv \Rightarrow P \propto t\) Hence, power–time graph is a straight line through origin.

Answer 1

The Correct Option is A

Step 1: Use Newton’s second law. Since the force applied by the motor is constant: \[ F = ma = \text{constant} \] Hence, the acceleration \(a\) of the body is constant.
Step 2: Write velocity as a function of time. For constant acceleration (starting from rest): \[ v = at \] Thus, velocity increases linearly with time.
Step 3: Write expression for power. Instantaneous power is given by: \[ P = Fv \] Since \(F\) is constant: \[ P \propto v \]
Step 4: Substitute \(v = at\). \[ P = F(at) = (Fa)t \] This shows: \[ P \propto t \] So, power increases linearly with time, starting from zero.
Step 5: Match with the given graphs.
(a) Straight line through origin — Correct
(b) Saturating curve — Incorrect
(c) Exponential-like curve — Incorrect
(d) Constant power — Incorrect Final Answer: \[ \boxed{\text{Option (a)}} \]