While out for a morning run, a woman jogger is momentarily distracted by an oncoming male runner wearing a T-shirt with a funny slogan. This disruption to her concentration causes her to run right into a tree at the speed of 12 km/h. Startled by her off-course maneuver, funny T-shirt man smacks into another oncoming male jogger who is running at 7 km/h while texting a friend on his Smartphone.
Both men are the same size and are moving at the same speed. In this collision course, who will end up with the most scrapes and bruises, the woman or the men?
You may assume the second scenario is worse, because the "relative velocity" of the two colliding men is double (7 km/h + 7km/h = 14 km/h), and thus greater than the woman jogger’s velocity with 12 km/h. However, this is a misconception.
The following thought experiment gives a better understanding:
In a safety test, at the same time, two cars run into both sides of a bridge pier. Since the bridge pier is very thick (high mass), the two cars do not notice each other's impact. Now, visualize the bridge piers getting thinner and consider whether anything changes. If both cars are the same weight and speed, the bridge piers can be as thin as desired. It does not have to withstand the power (more accurately the impulse), because on the other side, exactly the same power (the same impulse) acts.
The events in our riddle behave in a similar way. When the two men collide, in each case, only 7 km/h effects them and not—as initially assumed—the double speed of 14 km/h. Thus, the woman jogger sustains greater injuries from her unfortunate meeting with a tree at 12 km/h, as 12 km/h is greater than 7 km/h.