Expert: Daniel Mazur - 5/28/2009

Question
Two students were playing in a gymnasium, they were chasing each other. The boy was 6'4" and weighed 180lbs, the girl was 5'2" and weighed 120lbs. They were first jogging 5/m/s then increased their speed running at 10/m/s. They maintained 10/m/s for 5 seconds when unfortunately by accident they hit another student in the lower back. The other student was 6'2" and weighed 230lbs. The impact on the other student lasted 3/S. What was the force at impact? Explain with examples the impact/force significance ie. potential consequences for the third student as a result of that impact/force.

Answer
Chris,

I am not at all sure, if I get the problem questions right. If I understand it correctly, the grounds to the answer can be found in the beginning chapters (mechanics, conservation of energy and momentum) of any physics textbook. I guess one needs to get some practice in order to be confident in the physical qualitative reasoning.

The history of the jog and run of the couple is irrelevant for the consequences of the impact. Assume (not clear from the problem) that the couple are running together shoulder to shoulder - no idea, how this could be called "chasing each other", but only that makes sense for the physics problem. So to begin with you have a total mass of 300lb moving at 10 m/s just before impact.

Then you have the "victim" being hit - again I assume this means they ran into him or ran him over - who is 230lb heavy and as no information is given regarding his relative velocity with respect to the ground, I'll (again) assume that he is stationary (speed is 0 m/s) at the moment of impact. Hard to see, how such an accident could ever happen, but be it so.

Then we're told that he was hit in the lower back. This information translates scientifically that he was hit near his center of mass and therefore the qualitative CONSEQUENCES of the impact on the student are this: He will be pushed in the direction of the other people's motion, the impact will not knock him down nor lift him up either. If he is ligth on his feet (has quick reaction), he will be able to get into a run and avoid falling. This scenario would justify the very long (3 seconds) duration of impact. If he is, on the other hand, too much caught by surprise, the impact will couple with the his feet firmly on the ground and tip him over. This will probably mean that the other two will fall an top of him and then they'll roll together - although 3 seconds seem unrealistic.

For the impact FORCE VALUE, it is calculated as the ratio of the impulse transmitted (let me use the symbol DP) to the "victim" and of the time the impulse transmission took (those T=3 seconds). The DP is calculated thus:

We start with the equations (momentum conservation law)
(m1+m2)*v1 = (m1+m2+m3)*v2
the energy conservation law
(m1+m2)*v1^2 = (m1+m2+m3)*v2^2
and the impulse definition
DP = m3*v2

From this system we decide to eliminate the unknown v2. The intermediate equation is
(m1+m2)^3*v1^2 = (m1+m2+m3)*((m1+m2)*v1-Dp)^2

which after lengthy manipulation gives
Dp = v1*(m1+m3)*(1-sqrt((m1+m2)/(m1+m2+m3)))

So finally the (average) force will be
F = v1*(m1+m3)*(1-sqrt((m1+m2)/(m1+m2+m3)))/T = 337 kg*m/s / 3s = 112N

This is the mathematical solution. Personally, I don't like the problem much: it pretends to be "real life" problem, but bears many signs of being purely artificial.

Cheers!
Daniel