A velocity amplifier The three air-track carts shown in...

A velocity amplifier The three air-track carts shown in Figure $7.17$ have masses, from left to right, of $4 m, 2 m$, and $m$, respectively. The most massive cart has an initial speed of $v$; the other two carts are at rest initially. All three carts are equipped with spring bumpers that give elastic collisions. (a) Find the final speed of each cart. Notice that the smallest cart has a final speed greater than $v$. We refer to this effect as an amplification of the initial velocity. (b) Verify that the final kinetic energy of the system is equal to the initial kinetic energy. Assume that the air track is long enough to accommodate all collisions. (An example of velocity amplification is shown in Figure 7.18. Notice that the speed of the lightweight green ball is increased by its collision with the heavier orange ball, which sends it much higher than the original drop height.)

Key Concepts

Elastic Collision

An elastic collision is one in which objects collide without any loss of total kinetic energy. In such interactions, both energy and momentum are conserved, allowing the use of mathematical relationships to predict post-collision speeds.

Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a system remains constant if no external forces act on it. This concept is fundamental in analyzing collisions, as it provides a relationship between the masses and velocities of the colliding bodies before and after impact.

Conservation of Kinetic Energy

In elastic collisions, not only is momentum conserved, but the total kinetic energy of the system remains constant as well. This conservation law is crucial for determining the final velocities of the colliding bodies by adding an additional equation to solve for unknown variables.

Sequential Collision Dynamics

Sequential collision dynamics involves analyzing a series of collisions among multiple bodies. In such scenarios, energy and momentum are transferred step-by-step, sometimes leading to counterintuitive outcomes like velocity amplification, where a lighter object ends up moving faster than the initially moving heavy object.

Transcript

00:01 So this problem involves an elastic collision.

00:04 And with an elastic collision, we can apply not only the fact that, as always, momentum is conserved, and if there's no external forces, the momentum then should be constant before and after the collision.

00:17 But also, because it's elastic, we can assume that the kinetic energy at the beginning is equal to the kinetic energy at the end.

00:23 I say kinetic energy is restored.

00:28 So in the textbook, they derive a formative for this situation.

00:34 This is a one -dimensional collision where there's no external forces.

00:38 So they say it's taking place on an air track, just like in this problem.

00:41 And one of the objects, let's call it object 2, has initial velocity of 0.

00:47 And object 1 has initial velocity of v.

00:51 In that case, then, using the restoration of kinetic energy and momentum conservation, they derive in the text that the final velocity of object 1, which was the one that was moving initially, call that v1 final, is going to equal the mass of object 1 minus the mass of object 2 divided by the sum of the two masses, m1 plus m2, times initial velocity of object 1.

01:27 And then the final velocity of the second object, which was at rest initially, is going to equal just 2 times m1 divided by m1 plus m2.

01:47 And that's times the initial velocity of v.

01:52 So we can use those formulas to solve this question regarding these velocity amplifiers, it's called.

02:02 So let's first talk about the first collision there, when the mass of 4m collides with the mass of 2m.

02:10 Then in that situation there, the final velocity of the first mass, which turns out it's going to be its final, final velocity, because it's going to be moving slower than either of the other two things are going to be moving after their collision, is going to equal its mass, which is 4m minus the mass of object 2, which is 2m, divided by their combined mass, which is 4m plus 2m.

02:40 6m times the initial velocity, which is v.

02:45 And so this is 4 minus 2 is 2.

02:48 So you have 2m over 6m times v, which is equal to 1 third times v.

03:02 So that is the velocity of the first object after its collision.

03:08 Now, as we'll find out in just a minute, that's moving slower than the second object will.

03:11 So it's not going to catch up to it, obviously.

03:14 And so we can say that's going to be its velocity after all of these collisions happen.

03:22 The v2 final is going to equal, now, in this one it's just 2 times m1, which was 4m.

03:39 And then divided by the sum of m1 plus m2, well, this is 4m plus 2m.

03:44 So this is 6m again.

03:48 And we'll divide it out.

03:50 And we're looking at 8m divided or 8 divided by 6.

03:57 And this is times v, of course.

04:00 That gives us then 8 over 6 times v, which reduces to 4 thirds v.

04:12 But this is not the final velocity of this one, because the second object is, again, involved in a collision with the third object.

04:19 So we need to go ahead and calculate v2 final for real here.

04:25 This is v2 final, final, let's call it.

04:30 That's going to be then its mass of 2m minus the mass of object 3, which is just m, divided by the sum of the two, which is 2m plus m.

04:58 And this is times the object 2's initial velocity in this collision, which was its final velocity in the previous one, which was 4 thirds v.

05:10 And so that's going to come out to be 2m minus m is just m.

05:15 And then 2m plus m is 3m.

05:19 So that's m over 3m times 4 thirds v.

05:31 And so its final velocity is going to be then, m is cancelling out, 4 ninths v.

05:39 And so that's the second answer.

05:42 That's the final velocity of object 2 after both collisions.

05:46 Now object 3 after the collision, so velocity 3 final, is going to be, well, it's 2 times the object that was moving's mass.

06:02 So it's 2 times 2m.

06:07 And this is divided by 2m plus m.

06:17 But it's times the initial velocity of the second object again, as it goes into the third collision, which we saw already was 4v over 3.

06:29 So this is equal to 4m over 3m times 4 thirds v.

06:48 And so this is going to come out to be, the m's cancel out, we have 4 times 4 over 3 times 3 times v.

06:53 So this is 16 divided by 9v.

07:01 So that's the third answer.

07:03 So those are the three velocities that we have to establish...

Please give Ace some feedback