Momentum transfer from linear to angular motion

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The discussion centers on creating a momentum trap to measure the velocity of a projectile upon impact with a plate at the bottom of a pendulum. The setup includes a known mass plate and a pendulum of a specified radius, with the rotation angle recorded after the impact. The user believes their physics understanding is sound but is facing challenges with the mathematical calculations needed to convert the recorded angle into linear velocity. Assistance is sought to bridge the gap between the physical setup and the required mathematical translation. The conversation emphasizes the need for clarity in applying physics principles to practical scenarios.
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I am trying to create a momentum trap to calculate the velocity of a projectile when it hits the trap. It essentially consists of a plate if known mass at the bottom of a pendulum of known radius. When the projectile hits the plate, the degrees that the pendulum rotates are recorded. I believe my physics are correct, but I am struggling with the math to figure out how to translate those 3 pieces of information into linear velocity. Please help.
 
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For simple comparison, I think the same thought process can be followed as a block slides down a hill, - for block down hill, simple starting PE of mgh to final max KE 0.5mv^2 - comparing PE1 to max KE2 would result in finding the work friction did through the process. efficiency is just 100*KE2/PE1. If a mousetrap car travels along a flat surface, a starting PE of 0.5 k th^2 can be measured and maximum velocity of the car can also be measured. If energy efficiency is defined by...
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