What if we put the treadmill on a train, or in my example above what if one person was running in a moving room with a hole in the floor and someone else was running next to him on an actual treadmill?
And if you had frictionless roller skates and where pushed up to x MPH on a flat surface you would continue at that speed indefinitely (discounting wind resistance of course). And if you removed the roller skates you would just have to keep moving your feet under your already moving body.
I was at the gym today and I was shocked at how much cushion the treadmills have, defiantly a lot easier (impact wise) on your legs.
Only because the runner is running. If the runner stopped running and the treadmill was long the runner would appear to be moving backward at 10 kph (or whatever speed the treadmill was set to). To stop yourself from moving backward at 10 kph you must move yourself forward at 10 kph relative to the treadmill, this is exactly the same as moving yourself forward at 10 kph relative to the ground.
To expand on the train analogy, imagine the train is an empty freight train with flat carriages that allow someone to run steadily from car to car. The train is moving backward at 10 kph and a person is jogging forward on the train at 10 kph so that to an outside observer he appears to be remaining in place, just like a treadmill. Do you agree that if the runner slows down and stops running his entire body, feet, legs, torso and head will move backward at 10 kph? Not just his legs and feet? So in order to run on the train he needs to move all of his body, not just his feet and legs?
The analogy is not valid because wearing frictionless skates stops the runner from interacting with the treadmill belt. The runner must move his entire body because his feet are interacting with the treadmill and his feet are attached to his legs which are attached to his torso which is attached to his head. put him on frictionless skates and he may as well not be on the treadmill at all.
In the reference frame of the treadmill (which is all that matters) the runner is moving his body forward at 10 kph. If he did not do so his body would stay stationary reference the treadmill belt, it would not stay stationary reference the ground.
Lets put it another way. You say he only needs to move his feet and legs forward but not his torso. So if we had a long treadmill, not really long, lets say it is 10 metres long. So we set the treadmill to 10 kph and the runner runs along at 10 kph, staying in position relative to an outside observer but moving at 10 kph relative to the belt. If he only needs to move his feet and legs then if he stops moving his feet and legs what happens? Do they go backwards but his torso stays stationary until his torso is ripped from his legs in a mess of blood and guts? Or is it more reasonable that his torso remains attached to his feet and legs and therefore ends up going backward at 10 kph along with the treadmill belt? If his torso remains attached to his feet and legs then do you agree that in order to stop his body from moving backward along the treadmill belt he must not only carry his feet and legs forward but also his torso?
As I said earlier, this all assumes that the runner is not holding on to the hand grip at the front of the treadmill, if you do that then you are effectively getting a 10 kph tow from the arms and you would indeed only need to move your feet enough to stop them falling out from under you.
This is where our disagreement is, I think. If we give the runner motorized skates that can match the movement of the treadmill then the power of the skates only have to overcome the total friction between the runner and the treadmill. IOW the runner stays in the same space. The difference between our analogies might be that you’re assuming the runner starts after the treadmill is up to speed, I’m assuming the runner stays in place from the beginning.
To a person driving alongside someone running on the road, matching the vehicle speed to their pace, their torso is staying still and their legs are keeping time with the road, which is moving backwards. Linear motion/acceleration is relative - there is no privileged reference frame.
Ok, he’s got motorised skates. What happens if you turn the skates off? Does the entire runner, torso included, start moving backward with the belt, or is it only his legs and feet that move backward, his torso becoming separated in a sticky mess of blood and guts as it stays motionless relative to the ground? Assuming you can see that the skater is not going to get ripped apart and that his entire body will move backward with belt if the skates are turned off, can you then see that the motorised skates must move the entire runner forwards, not just his feet and legs? Can you see that this is exactly the same as if he was wearing those motorised skates on the ground (ignoring air resistance)?
One last try, this one is very similar to the motorised skates. The runner is actually driving a car on a large treadmill. A car is effectively the same as one very large motorised skate, it has an engine that turns the wheels and for a fuel consumption you get a given engine speed and for a given engine speed you get a given wheel speed. Lets say to drive the car at 10 kph the engine turns at 1000 revs which uses 1 litre of fuel per hour. If you drive the car on the treadmill at 10 kph the engine must turn at 1000 revs and it must use 1 litre of fuel per hour. If you then put the car on the ground and drive it a 10 kph (ignoring wind resistance again) the engine must still turn at 1000 rpm and it must still use 1 litre of fuel per hour. Fuel used is the same as energy, therefore you require exactly the same amount of energy to move the car forward at 10 kph on the treadmill or off the treadmill. If you then remove the car and leave only the runner, running along at 10 kph, he also needs the same amount of energy to run on the treadmill as he does to run on the ground (again, ignoring wind resistance.)
Running on the treadmill is the same as running on the ground, the difference between the two, the reason the treadmill is easier, is because the treadmill has an even surface, it has a lot more spring than hard tarmac allowing the runner to recover a certain amount of energy from each stride, and there is no wind resistance. The one thing that does not contribute to the treadmill being easier is that the runner only needs to move his legs and feet because his torso stays where it is all by itself.