04-Mar-2015: Non-constant acceleration problem/ Activity solution

Purpose: The purpose for the Non-constant acceleration lab was first, to solve the given non-constant acceleration problem analytically. And second, to show how instead of struggling with complex equations, that by using a spreadsheet, we can easily organize and calculate the answer numerically.

Procedure: We were given a problem involving the non-constant acceleration of a 5000 kg elephant on frictionless skates going 25 m/s down a hill and then on ground level. Then a rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephants direction of motion. Where the mass of the rocket changes with time so m(t) = 1500kg - 20kg/s*t. We wanted to know how far the elephant travels before coming to rest. Using Newtons second law and integration skills we aimed to derive the answer analytically.

Below is the analytical process required to get from the given variables to answering the question of how far the elephant traveled. Newton's second law gave us the acceleration of the elephant + the rocket as a function of time. Integrating the function from o to t to find change in velocity and then derive the equation for v(t).  Integrating that equation form 0 to t to find the change in x then deriving an equation for x(t) using that equation. After simplifying the equation using integration by parts we get a cleaner equation for x(t). In order to find the distance of the elephant we have to find t when the velocity reaches 0. Plugging in that time to the x(t) equation gave us x =248
.7 m. 

sideways image of problem setup and integration

This long, strung out process can be very time consuming and luckily for us we are able to do this problem numerically with the help of Microsoft Excel.

First we opened up a new Excel spreadsheet and made a series of columns such as change in t, a, a avg, change in v, v, v avg, change in x, and x. We began setting up the time increments by 0.1 for at least 220 rows. Using a change in t small enough was crucial in finding area under the curve by approximating a straight line. We assumed the function of the acceleration as 2t^2. Then we input a formula into cell B2 so that we can calculate the acceleration at any given time. Fill the formula down. Next in cell C3 we calculate the average acceleration for that first 0.1s interval. In cell D3 we calculated the change in velocity for that first time interval. (Using A3-A2 for the time rather than 0.1 s, so we could calculate this stuff using different time intervals if we chose.) In Cell E3 we calculated the speed at the end of that time interval. In cell F3 we calculated the position of the elephant (x at +v interval *change in t). we then filled down the data of row 3 to the rest of the spreadsheet in order to find when and where the elephant came to rest.

Results/Analysis: The spreadsheet gave us an answer that was extremely close to what we came up with analytically, proving that we can use a spreadsheet as a shortcut to solve messy problems. Two separate processes were used to solve the same problem and arrived at virtually the same answer.

Data table 

Conclusion/Uncertainty: This lab really showed us the logic of using a spread sheet to solve problems numerically rather than analytically in order to simplify a complicated process of integration and save time doing so. Plugging in simple equations into Excel spit out answers for an entire column of data. We were able to accomplish our goal of reaching the same conclusion using two separate processes and seeing how easy it is to use Excel to solve complex problems such as a non-constant acceleration.

A problem like this does not have much uncertainty because we are not physically measuring anything. The problem was to solve a complex problem both analytically and numerically using a spread sheet and comparing the results. Some error can occur in both processes but the coming up with similar results from both processes proved that an error, if any, was negligible.