Mar 2, 2015: Lab 3: Non-constant Acceleration

Purpose:

To use excel in a given non-constant acceleration problem to ultimately find the values of time and position; then compare it to the results done by integration by hand (analytical).

Data:
_ Analytical calculation (using integration):

• acceleration is a function of time and can be calculated through net force/mass of system, which are both given. Through Newton's second law, a(t)= F/m= (-8000N)/(5000+1500-20t)
• We have to integrate a(t) twice in order to get position as a function of time, x(t).
• We know that area under the curve of acceleration graph is the change in velocity,

Graphs of acceleration, velocity, position as functions of time. Vo=25m/sec

•  Solution from the lab handout gives the value, t= 19.69075sec, and x= 248.7m when the elephant comes to complete stop. (v-final = 0 m/sec)

__Numerical calculation (using excel)

        

       1.  Create the following cells in excel:

• a: function of time evaluated by intervals set by "t"
• t: manipulated time 
• a_average: average of points of "a" giving interval of time
• change in V: change in "a_ave" with given time interval
• V: function of time, an integral of a(t).
• V_ave: average of points of "v" given a time interval
• delta X: change in "V_ave" with given time interval
• X: adding changes in position delta X giving a running total.

         2.  Use time increment of 1second, note the highlighted row:

time increment= 1sec

Vfinal= -0.41~ 0m/sec; x=248.63m; t= 20sec

         3.    V-final= 0.12 ~0 m/sec; x= 248.69m; t=19.6sec

time increment= 0.1sec

         4.    V-final= -0.012m/sec; x= 248.698m; t= 19.70sec

0.05 increment

Conclusions:

1. Through 3 different numerical integrations, the results came out to be very close to answers from analytical solution.

2. With analytical solution to compare to, we can either take the same time and compare the difference between position in numerical and analytical or take the same position and compare the time. Without the analytical result, we are still able to find the position and time by looking at cell E (velocity). The value should be very close to zero since the elephant stops completely and that same row should have values for the position and time.

3. From this lab, we've learnt to use excel to do a shortcut to a rather long (by hand) integration problem. When comparing the answers from both ways, excel gives very reasonable values of position and time.