Question

If the dimensional formula of \[ (\text{Energy} \times \text{speed}) \] is \[ [M^aL^bT^c], \] then \(a,b,c\) are:

• A. \((1,3,-3)\)
• B. \((1,2,2)\)
• C. \((1,2,3)\)
• D. \((1,3,-2)\)

Hint:

When multiplying physical quantities, add the powers of corresponding dimensions.

Answer 1

The Correct Option is A

Step 1: Recall what Energy means physically.
Energy is the capacity to do work. Work is defined as force multiplied by displacement.
So, the dimension of Energy = dimension of Work = dimension of Force times Length.

Step 2: Write the dimension of Force.
By Newton's second law, Force = mass times acceleration.
\[ [F] = [M][LT^{-2}] = [MLT^{-2}] \]

Step 3: Write the dimension of Energy.
Energy = Force times displacement:
\[ [E] = [MLT^{-2}][L] = [ML^2T^{-2}] \]

Step 4: Write the dimension of Speed.
Speed = distance divided by time:
\[ [v] = \frac{[L]}{[T]} = [LT^{-1}] \]

Step 5: Find the dimension of (Energy times Speed).
\[ [E \times v] = [ML^2T^{-2}][LT^{-1}] = [ML^3T^{-3}] \]

Step 6: Read off the values of a, b, c.
Comparing with $[M^aL^bT^c]$:
$a = 1$, $b = 3$, $c = -3$.
\[ \boxed{(1,\, 3,\, -3)} \]