27-Sep-2014: Non-Constant acceleration problem/Activity: elephant

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.

Find how far the elephant goes before coming to rest.

Ideas: 

1. Newton’s 2nd law gives us the acceleration of the elephant + rocket system as a function of 

time: a(t)= Fnet

m(t)

= !8000 N

6500kg!20 kg

s t

= !400

325!t

m

s

2 

2. You can integrate the acceleration from 0 to t to find ∆v and then derive an equation for v(t): 

v(t)= v0 + a(t)dt.

0

t

! 

3. You can integrate the velocity from 0 to t to find ∆x and then derive an equation for x(t): 

x(t)= x0 + v(t)dt.

0

t

! 

4. You can solve v(t) to find the time at which v = 0. 

5. Then you can use the time you derived above in 4 and plug that into your expression for x(t) 

to find how far the elephant goes. 

Numerical integration: 

1. Open up a new Excel spreadsheet and enter the following:

2. Set things up so that the time increments by 0.1 seconds for at least 220 rows 

3. Input a formula into cell B2 that will let you calculate the acceleration at any time. 

4. Fill that formula down to cell B3. 

5. In cell C3 calculate the average acceleration for that first 0.1 s interval. 

6. In cell D3 calculate the change in velocity for that first time interval. (Use A3-A2 for the 

time rather than 0.1 s, so you can calculate this stuff using different time intervals if you so 

choose.) 7. In cell E3 calculate the speed at the end of that time interval.

8. In cell F3 calculate the position of the elephant. That will be 

xat the beginning of the interval +vinterval·!t. If you have done this correctly you should be able 

to “Fill Down” the contents of Row 3 to the rest of the spreadsheet and determine when and 

where the elephant comes to rest, AND the answers should agree closely with what you go 

doing things analytically. 

9. Change the time interval to 1 second instead of 0.1s and see if it makes a difference. 

10. Change the time interval to 0.05 s instead of 0.1s and see if it makes a difference

A2 enter 0
E2 enter 25
F2 enter 0
A3=A2+0.1
B2=-400/(30-A2)
C3=(B2+B3)/2
D3=400*(LN(325-A3)-LN(325-A2))
E3=E2+D3
F3=F2+E3*(A3-A2)
then fill down

find out the speed +0 to -0 betweeb 19.6s - 19.7s. And the elephant tarveld 247.4m during this time.