# Running on a treadmill (2)

However when I mention to him, that as you jump, you are being pulled backward, he disagrees vehemently. 'You are not oscillating back and forth, your head is not moving relative to the front of the treadmill'. 'You are able to push forward just enough to counteract the small push backward since your feet are only in contact for a small fraction of time'.

Since you have run some marathons, an interesting point comes up:

My roommate claims the following:
- he can run much faster on a treadmill than on a track for a given short distance.
- he likens this to running while someone is pulling the carpet from under your feet.
I think he is willing to demonstrate this if I ask, since all our theoretical discussions are going nowhere.
He has gone through great lengths to successfully convince family members, friends, engineers, whenever the topic comes up.
Since you are a runner, do you find you can run significantly faster on a treadmill?
Do you find different muscle soreness either?

I still maintain that when the two inertial frames of reference are (by definition) moving at a constant velocity relative to each other the two situations are absolutely equivalent.

However as the speed of the belt changes, if he wants to run faster, then there will be an acceleration – obviously. In this condition the above statement does not apply because the two frames of reference are not inertial – there is relative acceleration.
The movement of the legs of a runner on the treadmill or on the road is the same, however during the acceleration the runner on the road has to develop more power than the runner on the treadmill because the one on the road actually has to accelerate their MASS relative to the Earth and this is not true for the runner on the treadmill.

Over a short sprint quite a lot of the race is acceleration and so I would agree that here the two situations are not equivalent.

The two situations of constant velocity and acceleration must be kept quite separate. The pulling of the carpet is an acceleration case.

I agree about the time interval in jumping up but in order to jump at all you must be in contact with the belt or the ground for a certain length of time. You can't jump without bending your knees!
Ask him about running down the aisle in a moving train. It is all relative to the train. The fact that the train may be travelling at 100 m.p.h is not important to what happens inside the train.

The air is all important. Runners in the Olympics in Mexico City found it easier to break the sprint records because of the lower air density. Just think of Bob Beamon's worked record breaking long jump – that stood for years. I can imagine that the lack of the air running past does make a difference. As far as I remember my fluid mechanics the resistive force due to motion through a fluid is proportional to the velocity squared. Therefore as you run faster (as in the short sprint) the effects of this air drag become proportionately much greater.

There is one further point. In all this I have assumed that the runner has no effect on the treadmill – in other words the belt does not slip, and the treadmill does not move.

An interesting exercise that you might both like to think about is what happens if the runner runs on a treadmill that has a massless belt and rollers ands where the belt runs over the rollers with zero friction.

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