An air-track glider with mass m1 and speed v1 x i collides...

An air-track glider with mass $m_{1}$ and speed $v_{1 x i}$ collides perfectly inelastically with a stationary glider of mass $m_{2}$ (a) Show that the fraction of the initial kinetic energy remaining after the collision is $m_{1} /\left(m_{1}+m_{2}\right)$. (b) Explain why this result means that kinetic energy cannot be conserved in such a collision.

An air-track glider with mass $m_{1}$ and speed $v_{1 x i}$ collides perfectly inelastically with a stationary glider of mass $m_{2}$
(a) Show that the fraction of the initial kinetic energy remaining after the collision is $m_{1} /\left(m_{1}+m_{2}\right)$. (b) Explain why this result means that kinetic energy cannot be conserved in such a collision.

Key Concepts

Perfectly Inelastic Collisions

This type of collision occurs when the colliding objects stick together after impact. While momentum is conserved, kinetic energy is not conserved because part of it is transformed into other forms of energy, such as deformation, heat, or sound. This key concept underlies the analysis of energy loss during such collisions.

Conservation of Momentum

In any collision within an isolated system, the total momentum before and after the event remains constant. This principle allows one to calculate the resulting velocity of the combined mass in a perfectly inelastic collision, forming the basis for understanding how momentum is transferred even when kinetic energy is not conserved.

Kinetic Energy Transformation

During inelastic collisions, the kinetic energy present before the collision is partly converted into other forms of energy. The derived fraction of kinetic energy remaining after the collision illustrates that not all kinetic energy is retained, reinforcing the idea that while momentum is conserved, kinetic energy is diminished through energy transformation processes.

Transcript

00:01 All right, in this question we're talking about inelastic collisions.

00:04 We have an air track glider with mass m1, which is moving with velocity v initial, towards a stationary air track glider that has a mass m2.

00:13 The two collide completely inelastically, which means that they're going to stick together, and then they continue moving with some velocity i'll call v -sup f for v final.

00:22 So the question part a asks us to show that the fraction of kinetic energy after the collision versus before is equal to m1, over the sum of m1 and m2.

00:33 Let's start by defining the kinetic energy after and before the collision.

00:39 So we know that in general, k is equal to one -half mv squared.

00:44 So in this case, our initial kinetic energy is dependent only on mass m1, since m2 is not moving.

00:53 And our initial velocity was v -i.

00:57 Our final velocity, or excuse me, our final kinetic energy is one -half times the sum of their masses times the final velocity squared.

01:08 All right, so immediately we can take the ratio of these two by dividing.

01:16 So let's do that.

01:18 So the ratio is going to be equal to this term divided by this term.

01:24 So i'll rewrite this equation in terms of kinetic energy.

01:30 So k -final over k -i, or initial, is equal to the one -haves are going to cancel, so i'm not worry about that...

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