New data

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The ratio of momentum of cylinder to initial momentum of bob is rewritten as a function of length. The acceleration of the cylinder during the impulse is rewritten as a function of length

 

These equations were found using the lines of best fit from the new graphs found in the data analysis.

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The momentum equation can be rewritten  by using P=mv.

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Impulse​

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By using the equations for impulse, we are able to rewrite the momentum of the cylinder from the previous equation.

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Because we know that the pendulum bob-earth system only has gravitational potential energy at its highest point and only has kinetic energy at its lowest point, we can use conservation of energy in order to rewrite the initial momentum of the bob from the previous equation. Also, using trigonometry, we are able to rewrite height in terms of length of the pendulum.

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After rewriting the momentums, we can isolate time. The new equation represents how much time the impulse lasts depending on the length of the pendulum.

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The initial velocity of the bob before the impulse can be rewritten to ensure that the equation for time only depends on the length of the pendulum and physical constants. This new equation is very important in the computational model as it can be used to vary the duration of the impulse part of the simulation. This allows for a more realistic animation and smoother graph outputs.

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Velocity of cylinder

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Because we know the duration of the impulse and we know that the acceleration is constant during the impulse due to a constant net force, we can use kinematics to find the velocity of the cylinder by plugging in our previous equation for time and plugging in our acceleration function for acceleration.

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Energy

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We can also find the velocity of the cylinder by using energy.

 

By finding the mean of the percentage of total energy conserved in the data analysis, we can use energy equations to help isolate the velocity of the cylinder.

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Because we have an unknown variable, vb, we need to rewrite it using conservation of momentum. The momentum of the cylinder is replaced by the initial momentum of the bob multiplied by the momentum function which we found earlier.

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Velocity of cylinder

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By plugging in our equation for vb and replacing vb0 with one of the equations we found earlier, we are able to create an equation for the velocity of the cylinder using the energy method.

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Pre-impulse motion (code)

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These equations will help us model and update the position of the pendulum bob before the impulse.

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Impulse motion (code)

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These equations will help us model and update the position of the pendulum bob, position of the cylinder, and rotation of the cylinder during the impulse.

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Post-impulse motion (code)

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These equations will help us model and update the position of the pendulum bob, position of the cylinder, and rotation of the cylinder after the impulse. The value for acceleration of the cylinder was found in the data analysis by averaging the accelerations of the cylinder from the four trials.