Running Between Cars

Calculus Level 3

Two cars, 270 miles apart, start driving toward each other. One car has a speed of 40mph and the other has a speed of 50mph. Chris stands at the midpoint between the two cars. He runs toward one of the cars, touches its hood, then runs towards the other car, touches its hood, and repeats this until the cars collide (and presumably bounce off of him). Chris travels at a speed of 60mph at all times.

Let \(v(t)\) denote Chris’s velocity in mph and \(t_0\) denote the amount of time that has elapsed at the moment the cars collide (and presumably bounce off of Chris). Evaluate:

\[\left|\int_{0}^{t_0}v(t)dt\right| + \int_{0}^{t_0}\left|v(t)\right|dt\]

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