00:01
This problem actually consists of two problems, but we're going to solve both of them, so easy enough.
00:07
So the first problem is this one.
00:09
We have two carts, cart a, which has a mass of seven kilograms, and cart b, which has a mass of three kilograms.
00:16
And these two carts are held together, pushing together against the compressed spring, attached to one of them.
00:24
So when released these, the carts are pushed apart.
00:27
And by the way, we're going to assume that this spring is massed.
00:30
Because we can't really solve the problem unless we have a massless spring here.
00:35
Otherwise, we have to take them account the mass of the spring, which we don't want to do.
00:38
So when we release these carts, they're pushed to get part by the spring, and then cart b after release travels at six meters for second.
00:47
So the question is then how much potential energy was stored in the spring prior to it being released? now, we're going to solve the problem in the following way.
00:57
We're going to find the final speed of cart a.
01:00
And then we're going to use the speed of cart a and the speed of cart b along with their masses to find the kinetic energy of these two carts after they were released from the spring.
01:11
And that kinetic energy is equal to the potential energy that was stored in the spring.
01:16
That's where that kinetic energy came from.
01:18
So that's how we're going to solve the problem.
01:20
Now to find the the final speed of cart a, we just use the fact that momentum is conserved, which means since the initial momentum is zero because the two carts are just sitting there.
01:31
That means that after they're released, the momentum of cart a is equal to minus the momentum of cart b.
01:41
And so we can write this in terms of the masses and speeds as the mass of cart a times the speed of cart a is equal to minus the mass of cart b times the speed of cart b.
01:57
And so then we can solve this for the speed of cart a.
02:00
And so then we can solve this for the speed and the speed of cart a then is just equal to minus the mass of cart b divided by the mass of cart a times the speed of cart b.
02:12
And we know what the mass of cart a and cart b are and what the speed of cart b is, so we can then solve for the speed of cart a, which is equal to minus 2 .57 meters per second.
02:28
And that minuscine just sign just means that it's going in the opposite direction is cart b.
02:34
So i don't get freaked out about that.
02:37
That's nothing to worry about.
02:39
So then the kinetic energy we're going to find using the standard kinetic energy formula, right? the kinetic energy is equal to one -half mass of cart a times the square of the speed of cart a plus the math, one -half, the mass of cart b times the speed.
02:58
Of cart b squared and we put all those numbers in and when we do that we get that the kinetic energy have to release them is 77 .1 jules and that kinetic energy came from the potential energy contained in the spring so that means that the potential energy is equal to the kinetic energy and that's equal to 77 .1 tools.
03:25
So that's the solution to that problem.
03:28
Now, the next problem...