Two asteroids of equal mass in the asteroid belt between...

Two asteroids of equal mass in the asteroid belt between Mars and Jupiter collide with a glancing blow. Asteroid A, which was initially traveling at vA1 = 40.0 m/s with respect to an inertial frame in which asteroid B was at rest, is deflected 30.0 ° from its original direction, while asteroid B travels at 45.0 ° to the original direction of A, as shown in (Figure 1). Part A Find the speed of asteroid A after the collision. vA2 = m/s Part B Find the speed of asteroid B after the collision. vB2 = m/s Part C What fraction of the original kinetic energy of asteroid A dissipates during this collision?

Transcript

00:01 Okay, for this problem, it's basically a momentum conservation problem.

00:06 So initially, we have momentum just in the x direction, just moving to the right.

00:11 So that has to stay conserved.

00:13 And then we have none moving up and down.

00:16 So that means afterwards also the vertical has to kind of go in.

00:20 So to figure out these two velocities, what we're going to have to do is realize that mass times the initial velocity of a, that's our initial.

00:30 And this is the x direction, would have to equal, and since these all have the same mass, i'm just going to call them m.

00:38 So m times bfa in the x direction, plus m times bfa in, or, whoops, sorry about that, b, bfb in the x direction.

00:57 So to find those, what we have to do is kind of split these up.

01:01 So for example, we know that this would have a vfb x and then vfb y, and they would combine together to give us just vfb.

01:17 As i kind of look at this, what that means then is that vfb is the hypotenuse.

01:24 So it would equal vfb or actually vfb.

01:31 Is the adjacent side, so it would equal vfb times the cosine of 45 degrees, and vfb y would be vfb times the sign of 45 degrees.

01:49 And then the same thing kind of up here with our vfa.

01:54 Vfa would be the hypotenuse, so we'd have a vfax and a vfa so we can write those that vfax would equal vfa times the cosine of 30 degrees.

02:16 And then vfay would equal vfa times the sine of 30 degrees.

02:25 Now since cosine and sine of 45 are the same, these are actually going to end up being the same.

02:31 These are not going to be the same.

02:33 They will actually be different.

02:34 But we can calculate all of those.

02:36 So the other part we have down here is we also have a y direction.

02:41 And so we originally had zero y motion.

02:45 So what we have is we would have to have mass times the vfa in the y direction, and then it would have to actually e plus mass times the bfb in the y direction.

03:01 And since they both have the same mass, we can actually cancel those masses...

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