Elementary physics goes boink.

Classical Mechanics Level 2

Consider a one dimensional collision between two point masses with different masses $m_1$ and $m_2$ and initial velocities $v_1$ and $v_2$ . We define the ratio $r$ as the ratio of the difference in initial velocities to the difference in final velocities:

$r=\frac{|v_{1,i}-v_{2,i}|} {|v_{1,f}-v_{2,f}|}$ .

What is $r$ if the collision is perfectly elastic?

The answer is 1.

1 solution

David Mattingly Staff
May 13, 2014

In the center of mass frame, since the collision is perfectly elastic, the velocities just flip direction (from right-to-left to left-to-right, for example). So the magnitude of the velocity difference $|v_1-v_2|$ remains unchanged and $r=1$ .

Fun fact: In particle physics, there's also a ratio $x$ in collisions with a similar meaning to $r$ (the only difference is $r$ is concerned with velocities, while $x$ is concerned with momenta). This ratio $x$ is called the Bjorken variable of the collision.

When the Bjorken variable is equal to $1$ , the collision is perfectly elastic. When it is not equal to $1$ (usually, less than $1$ ), the collision is inelastic (or soft). The Bjorken variable tells experimentalists about the characteristics of the collisions between particles and in fact, from detailed calculations of $x$ , one can actually have some knowledge about the internal structure of the colliding objects! Indeed, the theory about the structure of proton (which is made of three quarks) has been verified by calculating $x$ and comparing with experiment.

Elementary physics goes boink.

The answer is 1.

1 solution

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