The three air carts shown in FIGURE 9-28 have masses, reading from left to right, of 4 m, 2 m, a

The three air carts shown in FIGURE 9-28 have masses,...

Key Concepts

Elastic Collisions

Elastic collisions are interactions in which both momentum and kinetic energy are conserved. This concept underpins much of collision analysis in physics and provides the basis for understanding how objects rebound off one another without losing energy to deformation, heat, or sound.

Conservation of Momentum

The conservation of momentum is a fundamental principle stating that within a closed system with no external forces, the total momentum remains constant before and after a collision. This principle is crucial in collision analysis as it provides a direct relation between the masses and velocities of the objects involved in the collision.

Conservation of Kinetic Energy

In an elastic collision, kinetic energy is conserved alongside momentum. This means that the sum of the kinetic energies of the objects before the collision is equal to the sum after the collision. This concept is pivotal for solving problems where the final velocities are unknown, as it allows the derivation of additional equations governing the system dynamics.

Sequential Collision Analysis

Sequential collision analysis involves breaking down a series of collisions into individual events. In each event, conservation of both momentum and kinetic energy can be applied independently to find the outcomes of each collision. The result from one collision becomes the input for the next, making it essential for understanding and solving multi-stage collision problems.

Relative Velocity Principle in Elastic Collisions

A unique property of elastic collisions is that the relative velocity of separation after the collision is equal in magnitude and opposite in direction to the relative velocity of approach before the collision. This principle provides an additional equation that helps in resolving the final velocities of objects after collision by linking the motion of the two colliding bodies.

Transcript

00:01 In question 40, we have three air carts that have masses 4m, 2m, and 1m.

00:05 The initial one has a velocity that's non -zero.

00:08 The other two have velocities that are zero.

00:10 The other two carts are at rest, and they're all going to give us perfectly elastic collisions.

00:15 They're one -dimensional collisions.

00:17 We're supposed to find the final speed of each cart, as well as the final kinetic energy of the system, and make sure it's equal to the initial kinetic energy of the system, which is aggravating.

00:28 We know that to be true, that's what elastic means.

00:31 But we're going to use two things that we know are true because it's a perfectly elastic head -on collision, which is that the difference in the velocities before the collision of two objects is equal to the opposite of the difference in the velocities after the collision.

00:47 So that is v1i minus v2i equals the opposite of v1f minus v2f.

00:53 I proved this in question 36.

00:55 I also know that the ratio of the velocity of the initial object from final to initial is equal to the ratio of the difference in the masses of the two objects to the sum of the masses of the two objects.

01:07 And i did prove this in question 39.

01:10 So we're going to go ahead and we're going to know that v1i equals v1i.

01:17 That's what we know.

01:17 We do not know v1f.

01:19 Over here, we know v2i is zero.

01:23 We are interested in finding v2f.

01:27 And over here, i'm going to call this v3.

01:30 We know v3i is zero.

01:33 We are interested in finding v3f.

01:36 So we'll start with this initial collision, and so i'm going to use this relationship first, v1f to v1i.

01:44 So v1f to v1i is equal to m1 minus m2, so that would be 4m minus 2.

01:55 M to m1 plus m2 that would be 4m plus 2m and so this is equal to 2m over 6m which is equal to 1 3rd so let me rewrite this i have v1f equals 1 3rd v1i i'm going to go ahead and hold on to that so that i can substitute it into this equation up here so i know v1i minus v2i but v2i is 0 is equal to the opposite of v1f v1f minus v2f which is actually what i'm interested in well v1 i is v1i that's what we're told equals so minus zero right but equals negative v1 f well v1 f is one third v1 i so negative one third v1 i plus and third v1 i plus because we distribute our negative v2f, which means i can add over this one -third and so i'm going to end up with four -thirds v1i equals v2f.

03:09 This final velocity v2f becomes this answer right here.

03:16 So we know after the first collision this thing is moving at four -thirds v1i.

03:22 However, the next thing that happens is another collision.

03:27 So this 2m is going to hit this 1m, and so i'm going to relabel this now as v2i initial equals 4 thirds v1i, and the reason i'm doing that is because this is initial as in before this next collision where these two things smash into each other.

03:48 So v2i equals four thirds v1i, and i do not know v2f because now i'm saying v2f, because now i'm saying v2f is the final velocity after this collision.

04:00 And so i'm going to use this equation.

04:02 I just got to remember my v1s and v2s are interchangeable...

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