9/4/2014 Non-Constant acceleration problem / activity

On this lab we went over a problem on non constant acceleration analytically and then used a computer to solve the same problem numerically. Here is the problem we solved:

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 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 the
m(t) = 1500 kg – 20 kg/s·t.

Analytically:

To solve this problem analytically we first had to use Newton's 2nd law to find the acceleration of the elephant plus the rocket system as a function of time. Here is the formula:

Then we integrated the acceleration from 0 to  t to find ▲v to derive an equation for v(t) as follows:

Next we integrated the velocity from 0 to t to find ▲x and then derive an equation fro x(t) as follows:

Solving v(t) to find the time at which v = 0 allows us to use that time to plug into the equation to solve for the distance of how far the elephant goes. Solving the equation analytically gives a distance of  248.7 m.

Numerically:

After we solved the problem analytically we used Excel to solve the problem numerically.

• We first created a row of time which incremented by 0.1 seconds for about 300 rows. 
• With the next column we input the formula to calculate acceleration at any time.
• In the third column we calculate the average velocity.
• The fourth row calculates the change in velocity
• The fifth row calculate the instance Velocity.
• The sixth row calculates change in distance.
• The last row calculate the instance distance.
• With all the columns going down to about 300 rows.

Here is a picture of a portion of the excel table:

 Here is the portion of the table where time is near 0 and as you can see we find a instance distance of about 248.69 m.

Conclusions:

1) To solve each problem takes a decent amount of time, yet if we needed to change any data it would be easier to find new results numerically. The results are almost the same even though the data on excel is not as close to zero as we could get we could always change the information to find a better result.

2) If you wanted to check if the time interval numerically is close enough to get a good result you could always change the interval of seconds on the first table to see how much the distance changes in out answer. We tested in class by decreasing the increment to see how much the answer would change and it did not change at all. If we did it the other way around and increased the time interval and noticed no new answer we can assume the time interval is sufficient