This laboratory is concerned with the question:

**If we drop an object from a distance 10,000 km
from the centre of the Earth, how long does it take
for it to fall to the surface of the Earth?**

There are four exercises to be done. You will need a straight edge of some kind (paper or cardboard OK) and a calculator to complete them.

In order to answer this question, we need to start with a few facts:

- the surface of the Earth is
*R = 6,370 km*from the centre; - the acceleration of gravity at the Earth's surface
is
*9.8 m/sec*^{2}; - the force of gravity is proportional to an object's mass and inversely proportional to the square of the distance from the Earth's centre.

Hence the force on an object at radius
* r ** mg/(r/R)*

Note that the sign is negative. If you drop an object, it falls. So
* r *

In this laboratory, we shall find
an approximate solution to this equation
by calculating a sequence of approximations to
position and velocity at times * 0** dt** 2 dt** dt ** n dt **r _{n}*

The position and velocity at subsequent times are computed approximately using the rules

: the differential equation determines the acceleration at radius*a*_{n}= -gR^{2}/r_{n}^{2} .*r*_{n} : this approximates the velocity by the constant*r*_{n+1}= r_{n}+ v_{n}dt throughout the time interval from*v*_{n} to*n dt* .*(n+1) dt**v*: this approximates the acceleration by the constant_{n+1}= v_{n}+ a_{n}dt throughout the time interval from*a*_{n} to*n dt* .*(n+1) dt*

These approximations should be better if * dt ***The problem this laboratory addresses is
how the accuracy of this approximation process
depends on the size of
the time interval.**

To run the process for a given inteval * dt ** dt ***Run** until the value of * t ***Step** through until the exact value is hit.

- Set
. Run and step through the process above to get approximate values for*dt = 0.5* ,*r(1263)* ,*r(1264)* .*r(1265)* - For each of these three values of
, record the pairs*t* on the graph and also in the table below.*(dt, r(t))* - Do the same for
and*dt = 0.25* .*dt = 0.125* - Use a straight edge to estimate what the value
of
would be if*r*_{approx}(t) were equal to*dt* . Plot the estimated points*0* ,*(0, r(1263))* ,*(0, r(1264))* , on the graph too.*(0, r(1265))* - Use a calculator to get more accurate estimates, and enter these new data in the following matrix as well.
- Keep in mind that what you would like to
do is set
*dt=0*or something very small, but that this is not practical. You must try to guess what would happen for small values of *dt*by using the data for large ones.

Answers in the table below must be as accurate as possible. You can do the plotting with the mouse, but to get the data in the first column below you will have to do some calculation.

Your Login ID is likely the first 8 characters of your name, in lower case.

Send mathematical questions about the labs to the professor in charge of Math 256: fournier@math.ubc.ca