Safer than Professor Rex's office hours
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When velocityraptor accelerates in the positive x-direction (i.e. when you hold down the right arrow key) the bullets moving in the negative y-direction (downwards) do some things that I don't understand in the frame of special relativity:
- The bullets' speed tends to increase and then decrease as velocityraptor's speed increases. In more detail: velocityraptor starts off with a speed of 0mph; as I push down the right arrow key her speed increases (the acceleration seems to be uniform) and the bullets' speed also increases; as she reaches 2.2mph (which seems to be about the same speed of the bullets when she is not moving) the bullets slow down.
- The bullets' shape increasingly shears along the vertical axis as velocityraptor's speed increases. In more detail: velocityraptor starts off with a speed of 0mph; as I push down the right arrow key her speed increases and the bullets' shapes get squashed; they get squashed more and more as her speed increases. At first I thought this was Terrell Rotation but this is still in the "measured" mode, not the "seen" mode.
What is the reasoning behind the above 2 phenomena? I have a feeling some math could help with the explanation so please, if you think it would be helpful, add some links/websites pointing to specific equations.
btw awesome game!
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Good questions - Sean - the bullets were far and away the most difficult addition to the game. What, physically, *should* happen is so counter-intuitive that I had a hard time making sure the right thing *was* happening.
For your specific questions, it looks like it boils down to Time. A couple levels will really help you play around with clocks: 18 and 44 (a 'bonus' level). Notice what all the clocks do as you run around.
Clocks you're running toward will appear more advanced that clocks you're running away from, with the size of the effect being greater the further they are from you. That solves your second point... when you're running toward the vertical line of bullets in level 8, the far side of the bullets is slightly more advanced in time than the near side, which skews them.
For the first point, it's a similar thing - as you're speeding up, the bullets ahead of you are advancing in time - once you stop accelerating, the ticking of their clocks is no longer 'accelerating' either.
Also, notice the bullets when you run *away* from them -- they'll appear to slow down, or even go backwards. Clocks you're running toward appear more advanced - and ones you're running away from appear behind.
Hope this helps!
Oh, and you asked for math - and thankfully this one is pretty straightforward. It just comes from the Lorentz Transformations, where time transforms as:
t' = gamma*(t - v*x/c^2)
So, put basically, the transformed time (t') depends linearly on displacement (x). To map this equation exactly on to what you're seeing happening with the clocks/bullets in Velocity Raptor, you'll need to be careful with signs and how you're defining your ref frames.