Wednesday, May 6, 2015

27-April-2015: Ballistic Pendulum

     The purpose of this experiment was to find the initial velocity of a ball by using the conservation of energy/momentum theories. This experiment was done by shooting a ball with some initial velocity into a container. The container had a hole in it so the ball would stay inside. The apparatus is shown below. The container is held up with string so it can move side to side.

     To the right of the container are tick marks for degrees. Also next to the container is a light rod that gets pushed out some distance by the ball and container. The rod is stiff enough that it does not fall back down to its original position. Doing this allowed for a reading of the angle the container made with the vertical at its furthest point after the ball collides with it. Below is a picture of the rod after the experiment and the angle tick marks.

     Since it was assumed momentum and energy were conserved, equations for both were produced. The experiment was split into two parts in order to make the equations work. The first part focused on the initial momentum of the ball and the final momentum of both the ball and container. The initial momentum of the container was neglected because it was initially at rest, therefore its velocity was zero and its momentum was zero. The equation for the conserved momentum was: 
     The second part of the experiment focused on conservation of energy after the collision. The ball and container were treated as one system. The collision gave the system some kinetic energy. Since the system moved to the right and up, the kinetic energy turned into gravitational potential energy and the system at its peak had zero kinetic energy. The equation for the conservation of energy was:
     The final velocity in the change of momentum equation is the same as the initial velocity in the conservation of energy equation. So, re-writing the momentum equation in terms of the ball's initial velocity and plugging that into the energy equation allowed for the solving of the ball's initial velocity. Below is the algebra needed to get to that final equation.

     We have the equation for the initial velocity of the ball being fired, but the work is not done yet. There was uncertainty in the experiment that needed to be accounted for. Measurements were made for the mass of the ball, the mass of the container, the length of the string holding the container up, and the angle the container made with the vertical at its highest point. So, a number for uncertainty was needed to give a range for the speed of the ball, and this was found with the equation for uncertainty. 
     The equation for uncertainty is quite a mouthful to describe in words, so below is the equation written out and each partial derivative used. In the top of the picture is the equation, in the blue are the partial derivatives, in the red are the measurements made, and in the bottom right in black are the uncertainties in those measurements.

     After letting the calculator figure out the rest, it returns a number for the initial velocity of the ball to be
 v=1.522 m/s (+-) .1479 m/s 

     This experiment was not just about getting a final answer to this problem. There is nothing to compare the answer to unless cameras and video analysis are involved, although the speed might make sense considering the ball is not shooting out of an actual gun. This experiment was also about how to approach problems differently. People are used to seeing a problem as a whole, but this problem required it to be broken down into two parts and use different theorems together.
     

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