00:01
Hello, everyone, today we're going to be talking about mass dilation. We've learned already about time dilation and length contraction. But mass dilation has the advantages of the fact that it's one of the events that we can regularly observe, and things like particle accelerators. And so, in these large machines that accelerate tiny particles to almost the speed of light, we can notice mass dilation as they move very quickly.
00:32
So how do we get to mass dilation Because momentum is a quantity that depends on velocity? And it depends on us. Right? So, it seems a good way to tie in. Now, we know that momentum is conserved in collisions, right? It turns out that even if we go down to a very, very small scale, and then look at individual atoms, we can still see momentum being conserved, no matter how fast they're moving.
01:02
So, we're going to assume for the momentum, that the conservation of momentum will always remain the same, even if we're travelling at relativistic speeds, right. Now, according to the principle of relativity, the laws of physics cannot change depending on how fast you're going. Right? You can't see a physical law, like conservation of momentum being violated just because you're moving fast.
01:31
So, what this means is that if we notice anything odd happenings in the conservation of momentum, we're going to have to change how we think about momentum. So, to demonstrate up a little example of how that momentum can change, we're going to be doing a thought experiment, except we're not going to be using a very fast-moving train, we're going to be using a collision between two spacecraft. Just because this experiment is a little hard to think of if we only use trains.
02:11
So, we have a short animation, showing us two spacecraft coming close to each other and colliding. Right. So, we say at the spacecraft are of equal mass, they're approaching each other at the same speed. Right, we can see the rebounding a little bit afterwards. Now, when they collide, we can see that they start moving away from our line in the middle. Right. So, it means that after the collision, each one of them has a component of motion in the y-direction. And of course, if we conserve momentum, then the component of motion will have to be the same for both, we can say that it's 10 meters per second. Right?
03:11
We can say that, that the collision doesn't affect their direction in the x-direction very much, right. So, we can say the velocity in the x-direction is about the same. Makes sense, we can see that the velocity in that direction is not really changing very much, especially if they only gained that amount of speed each way.
03:40
Now, let us suppose that instead of looking at it, from this point of view, where the ships are moving toward each other at equal speed, we look at it from the point of view of one of the ships. Now, this means that we are going to be starting to travel at relativistic speeds instead of being stationary. What this means, though, is that our definitions of meters and seconds are going to change. So, we will see what happens to the speeds going up and down, shall we?
04:23
So, let us go into the frame of reference of the bottom spaceship. So now as you can see, we start off as stationary. Now, if we look at what the first observer saw, when the spaceships collided, we know that after the collision, we are moving down at 10 meters per second. Right? This makes sense. We can see that the x velocity of this moving spacecraft is going to stay around us. So, we've got momentum conserved in this direction, starts off high to the right, ends up high to the right.
05:10
But how fast will this be moving upward? Even that we know about length contraction and time dilation. We don't need to worry about wind contraction. sure, we're moving upward, but we're moving upward at such a slow speed. Probably, that we don't need to worry too much about the distance changing. What we do you need to worry about is the time changing. Because now that we're inside this frame of reference, our seconds are going to be a different length to the first frame of reference.
05:48
So, if the first frame of reference says that this ship is going up at 10 meters per second, how fast is it actually going up? Well, we can use time dilation, right. So, if the observer is moving at 0.6 C, relative to us, then one of their seconds turns into 1.25 seconds. So, this ship isn't moving up at 10 meters per second. It's moving up at 10 meters per 1.25 seconds. And of course, 10 divided by 1.25 gives us eight meters per second.
06:38
Hello, now we have a problem. We've said the x momentum is conserved, right? Because that pretty much doesn't change. But now we've got two spaceships of equal mass. One's going down at 10 meters per second. One's going up at only eight meters per second. So, it looks like our momentum has changed. So how do we explain this if we still want to keep conservation of momentum? We can't change the velocity. But what else does momentum depend on? That's right, mass.