The physical motivation for the Laplace transform is causality.

Consider the linear input-output relation $$f_{\text{output}}(t)=\int_{0}^\infty R(t-t')f_{\text{input}}(t')\,dt'.$$ Causality dictates that the response function $R(t)$ is zero for $t<0$, to ensure that the output only depends on the input at earlier times.

Then if we know the Laplace transform $\hat{R}(s)$ of the response function (or transfer function), the output is related to the input by $$\hat{f}_{\text{output}}(s)=\hat{R}(s)\hat{f}_{\text{input}}(s).$$

The physical motivation for the Laplace transform is causality.

Consider the linear input-output relation $$f_{\text{output}}(t)=\int_{0}^\infty R(t-t')f_{\text{input}}(t')\,dt'.$$ Causality dictates that the response function $R(t)$ is zero for $t<0$, to ensure that the output only depends on the input at earlier times.

Then if we know the Laplace transform $\hat{R}(s)$ of the response function (or transfer function), the output is related to the input by $$\hat{f}_{\text{output}}(s)=\hat{R}(s)\hat{f}_{\text{input}}(s).$$ One could of course work instead with the Fourier transform, with the complication that the Fourier transform of a constant input only exists as a delta-function distribution (while the Laplace transform is simply $1/s$).

The physical motivation for the Laplace transform is causality.

Consider the linear input-output relation $$f_{\text{output}}(t)=\int_{-\infty}^\infty R(t-t')f_{\text{input}}(t')\,dt'.$$ Causality dictates that the response function $R(t)$ is zero for $t<0$, to ensure that the output only depends on the input at earlier times.

Then if we know the Laplace transform $\hat{R}(s)$ of the response function (or transfer function), the output is related to the input by $$\hat{f}_{\text{output}}(s)=\hat{R}(s)\hat{f}_{\text{input}}(s).$$

The physical motivation for the Laplace transform is causality.

Consider the linear input-output relation $$f_{\text{output}}(t)=\int_{0}^\infty R(t-t')f_{\text{input}}(t')\,dt'.$$ Causality dictates that the response function $R(t)$ is zero for $t<0$, to ensure that the output only depends on the input at earlier times.

Then if we know the Laplace transform $\hat{R}(s)$ of the response function (or transfer function), the output is related to the input by $$\hat{f}_{\text{output}}(s)=\hat{R}(s)\hat{f}_{\text{input}}(s).$$