Question

Arthur and Betty start walking toward each other when they are 100 m apart. Arthur has a speed of 3.0 m/s and Betty has a speed of 2.0 m/s. Their dog, Spot, starts by Arthur's side at the same time and runs back and forth between them at 5.0 m/s. By the time Arthur and Betty meet, what distance has Spot run?

Answer 1

In this problem, you can assume that Spot turns instantaneously as it doesn't give you any information about how spot's speed changes each time he turns around.

Knowing that, its easier to envision Spot simply running in a straight line, and stopping when Arthur and Betty meet.

We want to find the distance traveled by Spot. For anything traveling with a CONSTANT velocity, the distance traveled is simply distance = velocity x time

We know that spot runs with a velocity of 5m/s so that means time is our only unknown variable.

How do we find time? Well, we know that each person is walking towards the other (that gives us relative direction), we know each of their speeds, and we know the distance that they are traveling. To find the time, we need to know how long it takes until the distance between the two goes from 100m to 0m.

You can find time by simply dividing distance by velocity. You have to be careful to use the RELATIVE VELOCITY here. If only Arthur was moving, then the distance between them would decrease at a rate of 3m/s. If only Betty was moving, the distance would decrease by 2m/s. But since they are both moving TOWARDS each other, the distance between them actually decreases by 2+3=5m/s.

time = distance/velocity = 100m/5m/s = 20s

distance = time*velocity = 5m/s*20s = 100m

Its worth noting that, if Betty had been walking AWAY from Arthur, the relative velocity would be 3m/s+(-2)m/s=1m/s. Arthur would still close the gap, but much more slowly than when they walk towards each other. If Arthur is walking away, Betty is too slow and would not be able to catch up. That is reflected by the negative relative velocity (-3)+2=-1m/s that you would obtain in that scenario