$t=0$ and reaches destination after 1 hour. During the hike, his velocity in km/h varies according to the function $v(t) = \cos{\big(\frac{t\pi}{2}\big)}.$ Find the time in hours at which the hiker reaches the halfway distance between start and destination. Hint: $\sin{\big(\frac{\pi}{6}\big)} = \frac{1}{2}.$
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How does one compute the distance travelled between two time moments given an arbitrary equation for the velocity?
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… and if the time ranged between 0 and the generic time
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You should now have a generic equation that you can solve for time, knowing the previous total distance has now halved.
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The distance travelled in time
$\tau$can be calculated as follows:$$ \begin{aligned} \int_{0}^{\tau}{\cos{\Big(\frac{t\pi}{2}\Big)}} \,dt &= \left[\frac{2}{\pi}\sin{\Big(\frac{t\pi}{2}\Big)}\right]_0^\tau \\&= \frac{2}{\pi}\sin{\Big(\frac{\tau \pi}{2}\Big)} \end{aligned} $$Therefore, the distance travelled in an hour is$\frac{2}{\pi}\sin{\big(\frac{\pi}{2}\big)} = \frac{2}{\pi}.$Denoting the time in hours to travel half the distance as$x,$we get$\frac{2}{\pi}\sin{\big(\frac{x\pi}{2}\big)} = \frac{1}{2}\cdot\frac{2}{\pi}$which gives us$\sin{\big(\frac{x\pi}{2}\big)} = \frac{1}{2}.$Hence, using the hint, we can deduce$x = \frac{1}{3}.$