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If an arrow is shot from a bow which is of equal mass to the arrow and this takes place in space without anyone holding the bow, the bow and arrow should move apart in opposite directions at the same speed.
Now increase the mass of the bow without changing its power. If the bow weighs a thousand times as much as it did before, it will hardly move at all. The result of it staying almost stationary would appear to be that more energy will be transferred to the arrow because in the later stages of energy transfer from string to arrow the string will be moving forwards more quickly than on a bow which has begun to move backwards. In this case, the arrow must surely move faster and carry more energy than in the original case at the top, so the bow must move the opposite way with less energy.How is this compatible with the law that every action has an equal and opposite reaction?
The part I'm having difficulty with is that the bow releases a fixed amount of energy. In the case where bow and arrow have equal masses, half of that energy will end up in the movement of the bow and half in the movement of the arrow. If the bow has a much higher mass though, more than half of the energy appears to end up in the movement of the arrow, so less than half of the energy should end up in the movement of the bow. That does not seem to be equal.
Maybe the mistake I've been making is to think that the momentum of the arrow should be the same as the momentum of the bow, because maybe it shouldn't.
m1 v1 = m2 v2 = 0 ==> m1 v1 = -m2 v2Play around with that for a bit.
Momentum is always conserved. Action = Reaction even with a spring, just make an experiment and you will see... Don't think in terms of energy but momentum.
So the energy isn't balanced both ways at all. The same energy is being delivered from the spring in each case, but when there are equal masses at either side of the spring, the same amount of energy goes both ways. With unequal masses at either side, more energy goes into the lesser of the two masses, so it goes off with higher speed and higher momentum. If the momentum of the higher mass object is to match that of the lesser mass object, then the momentum of that will be higher too, but it doesn't matter that these values are both higher because all that counts with momentum is that they cancel each other out.
So the energy isn't balanced both ways at all.
A neat question! Once upon a time, it could have featured in an O level exam. Nowadays I guess you could get a degree for solving the equations.
No problem. The force is the same in both directions (Newton's third law) but the acceleration and hence the final velocity differs because the masses differ.
Late edit: Momentum isn't an abstraction! It's a property of anything that moves! It's the whole reason that aeroplanes and rockets fly, and the earth doesn't hurtle backwards when you walk! (it moves, but not a lot)
From the point of view of the 1kg mass, in one case it's accelerated to one speed (1kg mass at other end of the spring) and in the other case (2kg mass at the other end) it's accelerated to a higher speed in the same amount of time. That has to register as a higher force being applied to it.
What makes you think that? Nobody has mentioned time, and there's no reason to suppose that the acceleration time is the same for both experiments.
Try reading this.http://cnx.org/content/m42073/latest/