Given that $\dot{\theta} = 1$ for the linear segment, we can use the formula $\theta = \theta_0 + \dot{\theta}t$ to find the time taken for this segment.
Since $\theta_0 = 0$ and $\theta_l = 45^{\circ}$, we have $45 = 0 + 1t$.
Solving for $t$, we get $t = 45$
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