Verify that the collision between the proton and the nitrogen nucleus in Example 26.4 is elastic

Verify that the collision between the proton and the...

Key Concepts

Elastic Collision

An elastic collision is one in which the total kinetic energy and momentum of the system are conserved. In such collisions, no kinetic energy is lost to other forms of energy such as heat or deformation, making the analysis of the collision straightforward in terms of energy and momentum balances.

Conservation of Momentum

The conservation of momentum principle states that in a closed system, the total momentum before a collision is equal to the total momentum after the collision. This principle is fundamental in analyzing collisions since it applies regardless of whether the collision is elastic or inelastic.

Conservation of Kinetic Energy

In the context of elastic collisions, the conservation of kinetic energy implies that the sum of the kinetic energies of all objects involved remains constant throughout the collision. This is a critical concept for verifying elasticity, as any discrepancy in kinetic energy would signal an inelastic process.

Transcript

00:01 For this question, well, in order to prove, is it elastic or not? we need to prove if the initial kinetic energy is equal to the final kinetic energy.

00:09 Because if it's elastic, that means there's no net energy loss was happening during the process, which means that initial kinetic energy should be equal to final kinetic energies.

00:19 Well, first, let's take a look at the initial kinetic energy.

00:23 That's a k -i.

00:24 In this case, it's equal to gamma -p minus 1 times, mpc squared, okay? and since there's no kinetic energy on nitrogen atom, because nitrogen was at rest at the beginning, okay? so there was only proton was moving around at the beginning.

00:46 So therefore we have such equation here.

00:48 And gamma is a lorentz factor for proton.

00:51 And mp is the mass of proton, okay? and we know that the speed of a proton at the beginning was 0 .70c.

01:00 Therefore, we know that gamma p is equal to 1 over square root 1 minus vp square over c squared.

01:18 Okay? if we plug it back to the equation, we'll have initial kinetic energy is equal to 1 over square root 1 minus 0 .70 c and then square over c square and then um oh sorry and then minus one and then times mpc square okay and this will give us the initial kinetic energy is about 0 .4 mpc okay well and then let's look at the final kinetic energy in this case the final kinetic energy um we need to consider both protons and nitrogen atoms because in this case natural animal gain the speed after the collisions okay and if we set out equation we have say gamma p2 minus 1 times mpc square plus gamma n minus 1 times mn c square okay so the reason why is gamma p2 here is because after the collision, the speed of a proton was changed.

02:56 It was changing to wb2 was changing to 0 .63c.

03:07 Okay...

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