Partners: Matthew Ibarra, Billy Justin
Date of Lab: 12 September 2016
Lab 4 Non-Constant acceleration problem/activity
Mission Statement:To determine the distance covered by a rocket-equipped elephant on frictionless roller skates going down a hill, igniting the rocket the moment the elephant reaches level ground, which provides thrust in the direction opposite the motion of the elephant and whose mass is diminishing with time.
The exact phrasing of the problem is the following:
A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a hill and arrives on level ground. At that point a 1500-kg rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephant's direction of motion.
The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the mass of the rocket is depicted by the function m(t) = 1500 - 20t.
The exact phrasing of the problem is the following:
A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a hill and arrives on level ground. At that point a 1500-kg rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephant's direction of motion.
The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the mass of the rocket is depicted by the function m(t) = 1500 - 20t.
Theory: Previously, kinematic problems we've done were done under the precondition that the acceleration, if involved, be held constant. Now we must apply the principles of calculus in order to solve kinematic problems like the one described for this lab which involves non-constant acceleration. Furthermore, this problem also involves the mass of both the elephant and the rocket, with the mass of the rocket being a function of time, since fuel is being burned.
Experimental Procedure:
For this problem, we actually did not attempt the calculus-based method of solving the problem since the integration techniques required would have been too time-consuming for most of the class. Instead, we opted for a numerical approximation using Excel. For this problem, we knew the initial velocity of the elephant-rocket system, the (constant) mass of the elephant, the diminishing mass of the rocket, which was a function of time due to burning the fuel at a given rate, and lastly the (constant) thrust produced by the rocket when ignited.
Lists/Tables/Graphs of Collected Data with Explanation:
Upon opening the spreadsheet for Excel, we devoted columns to time, acceleration, average acceleration, change in velocity, speed, average speed, change in position, and the position of the elephant. Furthermore, we inputted the initial conditions provided, shown below, with the time changing in increments of 1.
We see that the 'v' column begins at 25 m/s but eventually diminishes to 0 m/s somewhere between
t = 19 and t = 20 seconds. During that time period, the distance traversed by the elephant before the rocket begins to reverse its motion is numerically shown to be 248.379 meters.
Next we changed the time increment size from 1 to 0.1 and obtained the data shown below.
The highlighted row is the row of interest. Now we see that when time is just before 19.7 seconds is when the speed of the elephant-rocket system is zero, and the distance traversed is 248.698 meters.
Finally, we changed the increment from 0.1 to 0.05 seconds and obtained the following data.
The relevant row here is highlighted and shows that when the time is somewhere between 19.65 and 19.7, the distance traversed is 248.698, which did not change from when the increment size has 0.1 and not 0.05, suggesting that calculus analytics would achieve the same number.
Experimental Procedure:
For this problem, we actually did not attempt the calculus-based method of solving the problem since the integration techniques required would have been too time-consuming for most of the class. Instead, we opted for a numerical approximation using Excel. For this problem, we knew the initial velocity of the elephant-rocket system, the (constant) mass of the elephant, the diminishing mass of the rocket, which was a function of time due to burning the fuel at a given rate, and lastly the (constant) thrust produced by the rocket when ignited.
Lists/Tables/Graphs of Collected Data with Explanation:
Upon opening the spreadsheet for Excel, we devoted columns to time, acceleration, average acceleration, change in velocity, speed, average speed, change in position, and the position of the elephant. Furthermore, we inputted the initial conditions provided, shown below, with the time changing in increments of 1.
t = 19 and t = 20 seconds. During that time period, the distance traversed by the elephant before the rocket begins to reverse its motion is numerically shown to be 248.379 meters.
Next we changed the time increment size from 1 to 0.1 and obtained the data shown below.
Finally, we changed the increment from 0.1 to 0.05 seconds and obtained the following data.
The relevant row here is highlighted and shows that when the time is somewhere between 19.65 and 19.7, the distance traversed is 248.698, which did not change from when the increment size has 0.1 and not 0.05, suggesting that calculus analytics would achieve the same number.
Conclusion- In fact, through calculus, we would have achieved the approximately same number of 248.7 meters, with the computational work shown on the first page of the lab, therefore confirming that our numerical technique is valid. Sources of error in this lab would likely only be due to human error during the calculations since there were no physically moving parts involved, but it would be very curious if there were.
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