Why Does E=mc2_ - Brian Cox [47]
Before the cannon is fired, there is no net momentum and the cannonball is sitting at rest inside the barrel of the cannon, which is itself standing still on top of a castle. When the cannon is fired, the cannonball shoots out at high speed, while the cannon itself recoils a bit but stays pretty much where it began, fortunately for the soldiers in the castle who fired it. The cannonball’s momentum is specified by its momentum vector, which is an arrow whose length is equal to the mass of the ball multiplied by its speed and whose direction points away from the cannon along the direction of flight as it emerges from the barrel. Momentum conservation tells us that the cannon itself must recoil with a momentum arrow that is exactly equal in length but opposite in direction to the arrow associated with the ball. But since the cannon is much heavier than the ball, the cannon recoils with much less speed. The heavier the cannon, the slower it recoils. So, big and slow things can have the same momentum as small and fast ones. Of course, both the cannon and the ball slow down eventually (and lose momentum as a result), and the ball changes its momentum because it is acted on by gravity. However, this does not mean that momentum conservation has gone wrong. If we could take account of the momentum taken by the air molecules that collide with the ball and the molecules inside the bearings of the cannon, and the fact that the momentum of the earth itself changes slightly as it interacts with the ball through gravity, then we would find that the total momentum of everything would be conserved. Physicists usually cannot keep track of where all of the momentum is going when things like friction and air resistance are present, and as a result the law of momentum conservation is usually applied only when external influences are not important. It is a slight weakening of the scope of the law, but it ought not to detract from its significance as a fundamental law of physics. That said, let’s see if we can finish our game of billiards, which is dragging on somewhat.
To simplify matters, imagine that frictional forces are completely removed so that all we have to think about are the colliding billiard balls. Our newfound law of momentum conservation is very valuable but it isn’t a panacea. It isn’t in fact possible for us to figure out the speed of the billiard balls after their collision knowing only that momentum is conserved and the masses and velocities of the balls before the collision. To be able to work this out, we need to make use of another very important conservation law.
We have introduced the ideas that moving things can be described by a momentum vector and that the sum of all momentum vectors remains constant for all time. Momentum is interesting to physicists precisely because it is conserved. It is important to be clear on this fact. If you don’t like the word “momentum,” then you could do much worse than to speak of “the arrow that is conserved.” Conserved quantities are, as we are beginning to discover, rather numerous and exceedingly useful in physics. Generally speaking, the more conservation laws you have at your disposal when tackling a problem, the easier it will be to find a solution. Of all the conservation laws, one stands out more than any other, because of its profound usefulness. Engineers, physicists and chemists uncovered it very slowly during the course of the seventeenth, eighteenth, and nineteenth centuries. We are speaking of the law of conservation of energy.
In the first instance, energy is an easier concept to grasp than momentum. Like momentum, things can have energy but, unlike momentum, energy has no direction. In that respect it is more like temperature,