2-March-2015: Lab 3 Non-Constant Acceleration

Lab 3: Non-Constant Acceleration Problem

Purpose: To solve a non-constant acceleration problem numerically and to learn essential skills on Excel along the way.

Problem: 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 a level ground. At that point a 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 m(t)= 1500kg - 20 kg/s*t.
Find how far the elephant goes before coming to rest.

Procedure : We solved this problem in two different ways.

In the first method, we gathered the given quantities in the problem and created an a(t) function. We then took the integral of a(t) to find the equation v(t). From here, we took the integral of v(t) to find an x(t) equation. In order to find how far the elephant goes before coming to rest, we set our v(t) = 0 an solved for t. Finally, we inputted the calculated value of the t into our x(t) equation and solved for x.

First Method: From the problem stated above, we gathered the following information:

• Elephant Mass = 5000kg
• Rocket Mass = m(t) = 1500kg - 20kg/s*t
• Mass of System = 5000+1500-20t = 6500kg - 20kg/s*t
• Initial Velocity (Vo) = 25 m/s

From here we create the acceleration function a(t):
a(t) = Fnet/m(t) = -8000N/ 6500kg - 20kg/s*t = -400/325 - t (m/s^2)

We then integrate the acceleration from 0 to t and then derive an equation for v(t):
v(t) = [25 - 400ln(325)] + 400ln(325 - t)

We then integrate the velocity from 0 to t and then derive an equation for x(t):
x(t) = [25-400ln(325)]t + 400[(t-325)*ln(325-t) - t + 325ln(325)]

To find how far the elephant travels before coming to rest, we set v(t)=0 and find that t = 19.69075(s)
We plug t into our x(t) equation and find that x = 248.7m
This means that the elephant traveled 248.7m before coming to rest.

The steps shown above are simply a summary and does not show each step required in order to solve the problem analytically.
For further explanation, please refer to the Lab 3 handout.

In the second method, we simply solved the problem numerically by using Excel. This method is easier and so much faster because solving the problem analytically is time consuming and creates a lot of room for error.

Second Method: We opened up a new Excel spreadsheet and set up 8 columns as seen below.

• First Column: Time
• Second Column: Acceleration
• Third Column: Average Acceleration over one time interval
• Fourth Column: Change in velocity for one time interval
• Fifth Column: Velocity
• Sixth Column: Average velocity over one time interval
• Seventh Column: Distance covered in one time interval
• Eight Column: Total distance covered at that time interval

Equations Used:

• First Column:  Time interval= .1 sec; small time interval = accuracy
• Second Column: a(t)= -400/325-t
• Third Column: a_av= a final- a initial/2
• Fourth Column: change in v= Change in a*t
• Fifth Column: V= Vo+change in v
• Sixth Column: Vav= Vf+Vo/2
• Seventh Column: Change in x = Vav*t
• Eight Column: Add change in x over each interval

In order to complete this process, the correct cell must be chosen and given an appropriate action.
For further explanation, please refer to the Lab 3 handout.

Note: All averages and change in velocity/distance occur over one time interval.

After we set up the columns with the appropriate equations, we then filled 220 rows by using Excel's fill down feature. It is essential that the equations in each column were filled properly, otherwise the data would not be accurate. If done properly the spreadsheet should look like this:

Now if you scroll down to the point where V = 0 m/s, t value and x value should be close if not the same to the values calculated in the first method.

To recap, we found that the elephant traveled 248.7m before coming rest and t=19.69075 sec.        (see first method)
In our spreadsheet, velocity is closest to zero between 19.6-19.8 sec. At these intervals, the distance traveled is 248.7m. As you can see, we found the same answer by solving the problem numerically.

Final Thoughts:

In this lab, we solved the problem in two different ways; analytically and numerically. Our results came out to be the same. We found that by solving the problem either analytically or numerically, the elephant still traveled 248.7m before coming to rest.

Choosing an appropriate time interval is essential in this lab. A large time interval can create a gap in the data and something too small would make the data difficult to analyze. In this case, we found that t=.1sec was appropriate through trial and error and also based on our analytic answer. If an analytical result is not available, you can use the numeric data and solve for the given information in the problem. In this case, I would use the numeric t-value and x-value and solve for Vo. The problem states that Vo=25m/s. So if my calculated value of Vo equaled the given Vo, I can conclude that the numeric data is accurate.