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Calculations with a Ballistic Pendulum

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In a ballistic pendulum an object of mass m is fired with an initial speed v_0 at a pendulum bob. The bob has a mass M, which is suspended by a rod of length L and negligible mass. After the collision, the pendulum and object stick together and swing to a maximum angular displacement theta as shown

a). Find an expression for v_0, the initial speed of the fired object. Express your answer in terms of some or all of the variables m, M, L, and theta and the acceleration due to gravity, g.

b). An experiment is done to compare the initial speed of bullets fired from different handguns: a 9 mm and a .44 caliber. The guns are fired into a 10 kg pendulum bob of length L. Assume that the 9 mm bullet has a mass of 6 g and the .44-caliber bullet has a mass of 12 g. If the 9 mm bullet causes the pendulum to swing to a maximum angular displacement of 4.3 degrees and the .44-caliber bullet causes a displacement of 10.1 degrees, find the ratio of the initial speed of the 9 mm bullet to the speed of the .44-caliber bullet, (v_0)_9/(v_0)_{44}

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