The impulse response is the system's reaction to an input impulse. In an RC circuit, the voltage source is the input, and the capacitor's voltage is the output. The system's state and output response before and after input excitation are distinctly defined.

Kirchhoff's law forms an input signal equation, with the capacitor's current and voltage providing the output. Substituting the current and dividing by RC yields a differential equation. The output for an impulse input is the impulse response.

The time constant τ=RC is introduced and the differential equation is multiplied by the integrating factor e^t/RC. Simplifying using the impulse function's sampling property and integrating within the system's limits results in:

This equation includes a step function and a dummy integration variable τ. Solving this equation gives the impulse response of the RC circuit. The graph of the impulse response shows an instant jump in capacitor voltage at t=0, highlighting the theoretical nature of a pure input impulse, as it is unrealizable in practical scenarios.

Understanding the impulse response is crucial for analyzing and predicting the behavior of linear systems. Knowing the response to an impulse, the response to any arbitrary input can be determined through convolution. This principle is fundamental in signal processing and control system design, where the impulse response provides essential insights into system dynamics.