Aaron
Max
Fergusson
Theory/Introduction: The purpose of this lab is to derive the distance an elephant wearing frictionless skates travels after skating down a hill onto ground level by using the computational power of excel instead of calculus. The reason we are using excel is because there are some problems where integrating with calculus would be impossible to do. In the scenario the elephant is wearing a 1500-kg rocket pointed the opposite way to slow down the elephant until it comes to a stop. Even though the rocket produces a constant force backwards, the rocket burns fuel as it slows down causing the weight of the elephant-rocket system to change which causes the system to experience a non-constant acceleration. We are given that the elephant has an initial velocity of 25 m/s at the bottom of the hill, the 1500 kg rocket produces a constant force of 8000 N, and the rocket burns fuel at a rate of 20 kg/s.
Apparatus/Experimental Procedure: For this lab we are first shown the analytical method to solve this problem.
The analytical approach makes an equation to find acceleration then begins to integrate it to get the equation for velocity. The velocity equation is then integrated to get an equation for distance. However, time had to be solved for since the position equation is a function of time. Time was solved by finding the time at which velocity had to be zero, and after plugging the time value into the position equation, the elephant was calculated to have traveled 248.7 m.
Instead of integrating several times, we can approach this problem numerically using the method of Riemann sums. Riemann sums let’s us take the approximation of integrals by taking the area under the curve.
Conceptually, we plot the acceleration as a function of time, then divide the time intervals into equal widths of Δt. The area under the curve is approximated by making trapezoids of width Δt under the curve. Each area of the trapezoids represents the average velocity value for its respective Δt which we can then make a new plot of velocity vs. time. We then take the area of the velocity vs. time graph using the trapezoid method which eventually gets us the distance the elephant traveled as shown in the excel spreadsheet.
The highlighted region is the time interval where the velocity changes from positive to negative. However, what we are looking for is when velocity equals 0 (when the elephant stops). This means that velocity = 0 between time interval 19 and 20. To get a more approximate value we cut the intervals between each time into a 1/10 of a second.
By cutting it into a 1/10 of a second we find that the elephant stops around t = 19.7s with a distance of 248.69 m.
Conclusion: When comparing our numerical results to the analytical results we found that the values were almost the exact same. Even without knowing the value for the analytical method we still knew our numerical result was very accurate since the change in time from 19.6-19.7s only yielded 7.1 millimeters in change. If we wanted to be even more accurate we could decrease the time interval to .001s. This excel method can get infinitely accurate results just by decreasing the time interval. To end the lab we were also tasked to determine how far an elephant would go if it weighs 5500 kg, the fuel burn rate is 40kg/s, and the thrust force is 13000 N.
By plugging these new variables into the excel, our spreadsheet calculated the new distance to be 163.1 m.
phys4AF18avnguyen 