## Question:

My roommate and I have had a long fierce debate on the following question:
Is running on a treadmill physically equivalent to running in a straight line on a stationary surface? I believe they are equivalent, while he believes it is an alternate workout, involving less running but more bouncing vertically.

PS. My roommate is currently working on his physics PhD, so I would love to prove him wrong!

Regretfully from the point of view of your room mate I have to agree with you. No matter what it may feel like (I have run on both treadmills and the road) they are both exactly equivalent.
Think of it like this.
First of all ignore the effects of air resistance (see later).
The two states are basically two inertial frames – whether the road is moving under you (treadmill) or you are moving over the road (normal case) the output of energy forma given speed is identical. Remember that on the road you have to apply a force in the forward direction to get you moving forwards while on the treadmill you have to apply a force ion the forward direction to stop yourself being moved back and falling off the back of the treadmill.
If you simply bounced up on the road you would land where you took off. Also if you bounced up on the treadmill you would land on the same spot on the moving belt from which you took off although this would be further "back" compared with the ground. This is because at the moment of take off you had a velocity towards the back of the treadmill.

Now to other effects.

The way in which you run on a treadmill might be different.
On the treadmill there is no air moving past you so you get hot and also the air around you might become "stale".
The only way in which the treadmill might be easier is that you do not have to run through the air as you do on a road run. I have done a few half marathons and the air resistance certainly makes a difference over those sorts of distances.

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