Two people have decided to use of a mountain trail to get some exercise. They start out from the parking lot at the bottom of the trail at the same time. Person 1 runs the trail at a constant speed v1 = 5 m/s. Person 2 walks the trail at a constant speed v2=2 m/s. Given that the people must return along the same path they climbed up, and given that the summit of the trail is d = 3 km from the parking lot, how far from the summit will the people be when they meet going in opposite directions? (Assume neither person pauses.)
Systems: Each person will be treated as a point particle.
Model: One-Dimensional Motion with Constant Velocity applies to each person separately. Depending upon how you visualize the problem, the model may have to be applied twice to the runner (person 1). We will explore this detail in the Approach.
Approach: This problem stretches the definition of One-Dimensional Motion with Constant Velocity. Even if we assume the path is perfectly straight, the runner must reverse direction at the summit, and so it would seem that person 1's velocity changes its mathematical sign within the problem. We will suggest two possible methods to deal with this issue.
Even though the dynamics of the motions described in this problem are very different if the path is curvy instead of straight, the kinematics are mathematically equivalent. It is mathematically possible to parameterize the motion along a non-self-intersecting path as a one-dimensional motion. Since this problem only deals with kinematics, our conclusions are valid for a curvy path as well.
Method 1
One way to be sure that each person has constant velocity in the problem is to shift the initial time.