02-Mar-2015 Non-constant Acceleration

PURPOSE: The purpose of this lab is to prove that the same solution to a problem can be derived numerically using an excel worksheet as the solution derived from solving it analytically, which is much more complicated and time-consuming.

PROCUDURE:

We were given a problem that asked for the final position of an object moving at a non-constant acceleration:

In class we went through the process of taking the acceleration function and integrating to get velocity as a function of time, and then integrating again to get position as a function of time.  After deriving these two equations and plugging everything, we found the time it takes for the elephant to come to rest, 19.69075 seconds, and then plugged that into our position function to see that the elephant goes 248.7 meters.

Numerical Procedure:


Using excel, we entered the following data:

• t               time(starting with increments of 0.1 but allowing us to experiment with different                       time segments later)
• a              acceleration
• a_avg      average acceleration
• Δv           change in velocity
• v              velocity (speed at the end of that time interval)
• Δx           change in position
• x              position at the end of that time interval

We first filled down time in increments of 0.1 seconds for 250 rows.  The equation for acceleration was the same acceleration as a function of time that we started out with when solving analytically: -400/(325-t).  This was plugged into B3 and filled down a few rows.  Next we found average acceleration for the first 0.1 second, which was (B3+B4)/2.  The change in time multiplied by average acceleration gave us change in velocity.  Velocity at the end of that time interval is the elephant's initial velocity, 25 m/s plus the change in velocity, which was E3+D4.  Average velocity was found similarly to how average acceleration was found, from the previous velocity plus velocity for current time interval divided by 2, or (E3+E4)/2.  Change in position was the change in time multiplied by average velocity.  Finally, position at the end of the time period was the previous position plus change in position: H3+G4.  Filling all the columns down for 250 rows gave us information for when velocity becomes zero.  The beginning of our chart gave us this information:

Numerical results with time interval of 0.1 seconds.

If we scroll down to when time is between 19.6 and 19.7, we see that velocity changes from positive to negative which means the elephant has come to rest and that is where we would end our data.  At this point of rest, the position shows a value between 248.6927 and 248.698, which is extremely close to our previous solution of 248.7 meters.

When we change the time interval from 0.1 seconds to 1 second, we don't get as close of a value to the position of the elephant when it comes to rest because it does not give us as close of a velocity to zero.  If you look at when time is between 19 and 20 seconds, velocity is 0.9 m/s and position is 248.3781 meters.  It is about 0.322 meters off, which is somewhat close but not as accurate as what we can achieve with a smaller time interval.

Numerical results with time interval of 1 second.

When the change in time is smaller, we create an area under the curve that is more precise between tx and tx+1 because the smaller the change in t, the closer the curve is to a straight line.  This creates a more approximate shape of a trapezoid, which is what we are calculating values from our acceleration vs. time or velocity vs. time graphs, as shown below for the change in velocity of an acceleration vs. time graph.

CONCLUSIONS:

1.  The results we achieve from doing the problem analytically and numerically are very similar.  Analytically, we got a result of 248.7 meters, and numerically with excel we got somewhere in between 248.693 and 248.698 meters.  This was effective and much simpler than spending time calculating it on paper.  

2.  If we didn't have the analytical result to which we could compare our numerical result, we would have to observe our results to see if there is a large gap in between data points.  If there is, we would have to analyze the results at smaller time intervals to derive more precise information.