Dimensionality check in dynamics: The unit of a system “time constant” (for first-order dynamics or analogous exponential responses) is the same as which of the following?

Introduction / Context:
Time constants appear in many engineering systems: thermal (lumped capacitance), electrical (RC), and mechanical (first-order lags). Knowing the correct units helps you interpret response speeds and compare different processes.

Given Data / Assumptions:

• First-order response form: y(t) = y_final + (y_initial − y_final) * exp(−t/τ).
• τ (tau) is the time constant.
• Standard SI base units are used.

Concept / Approach:
In the exponential term exp(−t/τ), the exponent must be dimensionless. Therefore, t and τ must share the same dimension. Since t is time, τ also has dimensions of time (e.g., seconds). This reasoning is fundamental and independent of the particular physical domain.

Step-by-Step Solution:

Verification / Alternative check:
For an RC circuit, τ = R * C; units are ohm * farad = second, confirming the time unit.

Why Other Options Are Wrong:

• Velocity, length: Unrelated dimensions.
• 1 / Time: That is angular frequency or decay rate, not the time constant.
• Dimensionless gain: Different concept (static gain).

Common Pitfalls:
Mixing up time constant τ with its reciprocal (decay rate 1/τ) used in some transfer functions.

Final Answer:
Time