Kinetic Energy Problem???

[Mat_D]

You are sitting at the side of a railway track, armed with a laser pointer velocity measuring gun.  To accurately measure the velocity of any moving object, you just point the laser at it and pull the trigger.

A train comes past and you spot a friend of yours sitting in a seat.  You point the laser at him and measure his velocity.  It tells you it is 10m/s.  You conclude that the train is travelling at 10m/s. (1)

Now, the train track is a large loop and soon the train comes past again at the same velocity, but this time your friend is running down the train in the direction of motion.  You measure his velocity and you get a reading of 12m/s. You conclude that the train is travelling at 10m/s and he is running at 2m/s down the train. (2)

You then work out his Kinetic Energy in both cases (1) and (2)

KE = � * Mass * Velocity Squared.

(You know your friend weighs 100 kg)

(1) When he is sitting;

KE = � * 100 * (10 * 10) = 5000 Joules

(2) When he is running;

KE = � * 100 * (12 * 12) = 7200 Joules

Therefore by running he has increased his kinetic energy by 2200 Joules. (7200 � 5000)

Now, imagine the same thing happens but this time the train is travelling twice as fast, at 20m/s.

(3) When he is sitting;

KE = � * 100 * (20 * 20) = 20000 Joules

(4) When he is running at 2m/s (combined velocity is 22m/s);

KE = � * 100 * (22 * 22) = 24200 Joules

This time, by running, he has increased his kinetic energy by 4200 Joules. (24200 � 20000)

This would suggest that the faster the train is moving, the harder it is to move, run or walk down the train! (as you have to put more energy into moving and therefore do more work)  This simply cannot be right, as people on a supersonic jet would be unable to move! 

What is going on?

[Mat_D]

'you can calculate energy in any inertial frame of reference you want, as long as you keep it the same in all your calculations.'

And that's the exact 'trick' employed in this mathematical/scientific puzzle Space!

The frame of reference jumps between the observer and the man on the train (whose frame of reference is the moving train).  This is why the equations fail.

It's a bit like those mathematical tricks that have a hidden 'divide by zero' and give impossible results.

Simple but clever isn't it?

[Space]

Sorry to make multiple posts...
If we decide to use the train as the reference, the problem seems to be more simple, but don't forget:
Because of the conservation of momentum, if the man accelerates in one direction, then something has to accelerate in the opposite direction. If it's not the train, it's the ground. The ground is already moving towards the rear of the train, and accelerates in that same direction.
Which means: earth's speed , relative to the train , slightly increases. Where does that energy come from? We end up with pretty much the same problem.

[Space]

I see what you mean, Mat_D, but I think that the amount of energy spent by the man 's muscles are the same in any inertial frame of reference.
It depends on the amount of sugar ( or whatever molecules ) consumed by the muscles, and that doenst change when you jump to another reference...

[Mat_D]

You are, of course, right.  The amount of energy used by the passenger  running in the two cases obviously does not change.  The question implies that by saying 'this simply cannot be right'.

But, using the KE equations, the observer gets unsettling results that can only be interpreted in the way stated - i.e. the passenger does more work in the second case - but he knows this must be wrong!

The question was WHY does he get these results.

From the observers point of view, he knows only the passenger has changed state, and therefore is the only body to add energy to the system.  Therefore his conclusion must hold.  His mistake is switching between one frame of reference making the calcs (his) and interpreting them in another frame of reference (the trains)

[Space]

That sentence
" he knows only the passenger has changed state, and therefore is the only body to add energy to the system "
doesn't make any sense.
You don't need to "change state" to add energy to the system.
I don't know what you call "to change state", but if it's keeping a constant speed and mass, it's definately not a reason to think it doesn't add energy.
If in that "state" you include fuel level, then you can't say it doesn't change state, you don't know whether it does or not.
You should try to justify your statement before basing your calculations on it. It is the weak link in your reasoning but you seem happy with not justifying it and using it anyway.

The calculations made in the reference of the observer are correct. The interpretation "it seems to takes more energy when the train moves faster" is valid in the observer's reference, and there is something wrong

[Space]

(oops, pressed the button accidentally)
... and there is something wrong there, even if you stay in the observer's frame of reference. It should take just as much energy whatever the train speed, in the observer's reference.
Staying in the observer's frame of reference doesnt remove the problem.

[Mat_D]

I do know what you are saying, but this problem really is not about the train.  Forget the train.  The train doesn't really need to exist.  It's enough to say for the purposes of this problem that the passenger is in a frame of reference that is moving with respect to the observer, and the observer makking correct observation and calculation.  It's his results that dont make sense to him.  Forget the food the passenger has eaten.  Forget the fuel the train is consuming. Forget equal and opposite reactions.  Forget friction.  Forget the world rotating. Just think about KE=1/2m(v*v) in the purest classical sense. This problem is about a simple GSCE level physics equation and uses a 'trick' to make it fail.

Would have been great to have asked this question in a school physics lesson.....

[Space]

Well if you want to give up and forget the problem, then that's fine with me, I was starting to find it a bit boring to keep pointing at errors that don't seem to matter to you. Maybe none of it really matters after all.

That simple equation in itself doesnt "fall" by any trick. The reason it fell in your particular problem is ... forgotten.

Maybe you just wanted to talk about the surprising fact that, depending on which frame of reference you are, the kinetic energy has a completely different behaviour ?
You can even find cases where it increases in a frame of reference while it decreases in an other one.
But as long as you use inertial frames of reference, energy is being conserved, and your source(s) of mecanical energy (if any) provide just the same amount of power, although that power is being distributed in different ways depending on the reference.