Question

For a given mass m, linear velocity v, linear displacement s, linear acceleration a and time t, the dimensionally INCORRECT expression for power P is

• A. \(\frac{1}{2} mav\)
• B. \(\frac{mv^2}{t}\)
• C. \(\frac{1}{2} mv^2\)
• D. \(ma^2 t\)

Hint:

Power is Energy/Time. Always check if the expression looks like a known energy term divided by time. Since \(\frac{1}{2}mv^2\) is exactly the formula for kinetic energy, it cannot also be the formula for power.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Power is the rate at which work is done or energy is transferred.
Dimensional formula for Power:
\[ [P] = \frac{[Work]}{[Time]} = \frac{[Force][Displacement]}{[Time]} = \frac{[MLT^{-2}][L]}{[T]} = [ML^2T^{-3}] \]

Step 2: Detailed Explanation:
Let's check the dimensions of each option:
(A) \(mav \implies [M][LT^{-2}][LT^{-1}] = [ML^2T^{-3}]\). (Correct)
(B) \(mv^2/t \implies [M][LT^{-1}]^2 / [T] = [ML^2T^{-2}] / [T] = [ML^2T^{-3}]\). (Correct)
(C) \(\frac{1}{2} mv^2 \implies [M][LT^{-1}]^2 = [ML^2T^{-2}]\). This has dimensions of Energy (Kinetic Energy), not Power. (Incorrect)
(D) \(ma^2t \implies [M][LT^{-2}]^2[T] = [M][L^2T^{-4}][T] = [ML^2T^{-3}]\). (Correct)
(E) \(mas/t \implies [M][LT^{-2}][L]/[T] = [ML^2T^{-3}]\). (Correct)

Step 3: Final Answer:
The expression \(\frac{1}{2} mv^2\) is dimensionally incorrect for power.