Phys Lab 2

Phys Lab 2 - MeasuringtheAccelerationofaFreelyFallingBody...

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Measuring the Acceleration of a Freely Falling Body and a Bungee Jumper John Domanick 9/21/10 Partners: Jordan Pardo, Matt Weiner Teaching Fellow: Tim Harden
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Abstract: In this experiment we investigated the motions of both a steel ball in free fall, and a bungee jumper on an elastic cord. For the steel ball we investigated the acceleration due to gravity, g. Theoretically we know that the value of g is 9.8m/s2, and we hoped to determine how well this value agrees with our experimentally determined value. We know that position, velocity, and acceleration of an object are all related through differentiation, that is to say, that velocity function of time is the first derivative of the position function of time, and that the acceleration function of time is the derivative of the velocity function. By measuring the position for the steel ball as it fell and differentiating the position values, and then plotting the velocity against time, we found g to be 10.5±0.1 m/s 2 . For the second part of the experiment, we investigated the oscillating motion of a bungee jumper on an elastic cord. We were able to measure the position as a function of time, and then determined both the time functions for velocity and acceleration as the bungee jumper moved up and down in motion.
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Theory: During a fall, the falling body accelerates due to the force of gravity. The magnitude of this acceleration is g, which is accepted to be equal to ‐9.8 m/s 2 . There are multiple methods of experimentally determining the value of g. The first is to simply plot y‐position vs. time and fit a quadratic equation (ax2+bx+c) to the data. This gives a value for g due to the equation for position during single line motion: y = 1 2 gt 2 + v y o + y o (1) Where y equals the position of the body, t equals time and v equals to velocity. The “a” coefficient of the quadratic fit is equal to half of g. Due to the fact that this method relies on a quadratic fit of data, the value for g obtained is only as good as the fit is to the data. A poor fit will give a poor g. There is a second method to determine g, which is to plot the velocity against time, and fit a linear model (mx+b) to the data, which gives a value of g due to the equation for velocity during single line motion: v y = gt + v y o (2) Since we can’t measure the body’s velocity as it falls, we need to be able to somehow calculate it, which we can do by taking the derivative of the body’s position data. If r(t) represents a body’s position at time, t, then the velocity is: v ( t ) = dr dt = y ' (3) With the data we collect in lab, we cannot simply take the functional derivative of position though, as we only have positions for discrete values of t, so we must estimate v(t), which we can do using the Taylor series expansion of r(t) about t, so that: v ( t ) r ( t i + Δ t ) r ( t i −Δ t ) 2 Δ t (4) We can also find g by finding the acceleration in the y direction, since we know that gravity is the only force acting on the body in free fall.
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This note was uploaded on 10/06/2011 for the course PHYS 19 taught by Professor Blocker during the Fall '10 term at Brandeis.

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Phys Lab 2 - MeasuringtheAccelerationofaFreelyFallingBody...

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