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The Toy Story Theorem: Ep. 2

pegkowalski

845 views

Let's bounce right in.

An elastic equation is an encounter between two objects in which the kinetic energy is conserved from before the objects hit to after.

Lucky for us, it seems apparent that Buzz performs a perfect elastic collision! Tens across the board! No splash. We're lucky because that saves us a lot of work.

In an imperfect elastic equation, where the collision may be at an angle and not head on, sometimes we end up with a different velocity for each of the objects after the collision. Buzz however, manages to bounce on a giant (classic) bouncy ball and come back without losing velocity and without moving the bouncy ball in the slightest...

Sure, Pixar may have some flawed fzx concepts there. Sue them.

Actually don't because they do absolutely INCREDIBLE work.

That was a pretty good pun, give me some credit! I'm moving UP in the world.

Moving on; let's pretend that the bouncy ball does indeed NOT absorb any of the energy and move itself, and Buzz gets it all back. In this case:

m1 (v1) = -m1 (v2')

In reality, the necessary equation may look a little more like this:

m1 (v1) + m2 (v2) = m1 (v1') + m2 (v2')

Whereas:

m1 = the mass of Buzz Lightyear

m2 = the mass of the bouncy ball

v1 = Buzz Lightyear's initial velocity

v2 = the bouncy ball's initial velocity

v1' = Buzz Lightyear's ending velocity

v2' = the bouncy ball's ending velocity

Again, though - ACCORDING TO THE MOVIE - we don't have to do that much work.

And because Buzz is always the same mass, the masses will cancel out once again. So all we will need is his initial velocity and we will then know, that his velocity hitting the bouncy ball is equal to that as he bounces away from it, except in the opposite direction (hence the negative sign).

v1 = -v1'

In my last blog we found his velocity exiting the hot wheels tracks to be 2.426 m/s.

To find Buzz's v1 we will have to do one simple kinematics equation. Buzz takes the time to fall another 0.5m before hitting the bouncy ball, so we have to account for that. Alas, the acceleration in the x-direction is 0 m/(s^2) so to find the velocity we are looking for:

Vf = Vo + at

But we can cancel out the second term! So:

Vf = 2.426 m/s

Buzz hits the bouncy ball with the same velocity in the x-direction that he left the hot wheels track with. Now we can find his velocity leaving the bouncy ball which, as we have already established, is the same. Just in the opposite direction! So we change the sign:

v1' = -2.426 m/s

I know this seems complicated, and mostly hypothetical and Rex may be screaming "I DON'T WANNA USE MY HEAD!" and that's okay. Because no matter what: Toy Story still exists.

Stay tuned for what happens next in Buzz Lightyear's exciting flight as he grabs the fan and we explore the great phenomenon of centripetal acceleration...ooohhh!!!



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