To solve the Laplace transform of $f(t)$ as defined in the problem:

1. Function Definition: \begin{cases} \frac{t}{\tau}, & 0 < t < \tau \\ 1, & t > \tau \end{cases}$$

Step 1: Laplace Transform Formula

The Laplace transform of $f(t)$ is defined as: $\mathcal{L}[f(t)] = \int_{0}^{\infty} f(t) e^{-st} \, dt$

We'll split the integral into two parts:

1. $0 < t < \tau$ where $f(t) = \frac{t}{\tau}$,
2. $t > \tau$ where $f(t) = 1$.

Step 2: Compute the Integral for $0 < t < \tau$

$\mathcal{L}_1 = \int_{0}^{\tau} \frac{t}{\tau} e^{-st} \, dt$ Let $u = t$, $dv = e^{-st} dt$, using integration by parts: