The dimensional formula for impulse is same as the dimensional formula for

• A. Momentum
• B. Force
• C. Rate of change of momentum
• D. Torque

To determine which option has the same dimensional formula as impulse, we first need to understand what impulse is and how it relates to other physical quantities. Impulse is defined as the change in momentum of an object when a force is applied over a period of time. Mathematically, impulse (J) can be expressed as:

J = F × Δt

Where F is the force applied and Δt is the time duration over which the force acts. Now, let's break down the dimensions of impulse and compare them with the options provided.

Dimensional Analysis of Impulse

The dimensional formula for force (F) is derived from Newton's second law, which states that force equals mass times acceleration:

F = m × a

Here, the dimensional formula for mass (m) is [M], and for acceleration (a), which is the change in velocity over time, it is [L][T^-2]. Therefore, the dimensional formula for force is:

[F] = [M][L][T^-2]

Now, substituting this into the impulse formula:

J = F × Δt = [M][L][T^-2] × [T] = [M][L][T^-1]

This shows that the dimensional formula for impulse is [M][L][T^-1].

Comparing with Other Quantities

• A. Momentum: The dimensional formula for momentum (p) is given by:

p = m × v

Where v (velocity) has the dimensions [L][T^-1]. Thus, the dimensional formula for momentum is:

[p] = [M][L][T^-1]

• B. Force: As calculated earlier, the dimensional formula for force is:

[F] = [M][L][T^-2]

• C. Rate of change of momentum: The rate of change of momentum is defined as:

Rate of change of momentum = Δp/Δt

Since Δp has the dimensions of momentum, we have:

[Rate of change of momentum] = [M][L][T^-1]/[T] = [M][L][T^-2]

• D. Torque: Torque (τ) is defined as the product of force and the distance from the pivot point:

τ = F × r

Where r is the distance (with dimensions [L]). Thus, the dimensional formula for torque is:

[τ] = [M][L][T^-2] × [L] = [M][L²][T^-2]

Final Comparison

Now, let’s summarize the dimensional formulas we derived:

• Impulse: [M][L][T^-1]
• Momentum: [M][L][T^-1]
• Force: [M][L][T^-2]
• Rate of change of momentum: [M][L][T^-2]
• Torque: [M][L²][T^-2]

From this analysis, we can see that the dimensional formula for impulse is the same as that for momentum. Therefore, the correct answer to your question is:

A. Momentum