Problem 3

In this problem we are trying to study the muzzle 
velocity and air resistance of a cannonball.  We use 
a combination of a still camera and a stroboscopic 
light with 1/500 sec between pulses.  On the picture 
we get a series of images of the cannonball taken 
exactly 1/500 sec apart.  We know that the position 
of the cannonball will be quadratic in time (at least 
for short times) since it slows down due to air 
resistance. Note that the effect of air resistance at
high velocities is far greater than that of gravity!
The equation describing the motion is thus:

b = U (t - t0) - 0.5 * a * (t-t0)^2

where we have as the three parameters the initial 
velocity U, the acceleration (actually the 
deceleration) a, and the initial time to leave the 
muzzle t0.  We lack information on the precise 
time it exits because of our measurement approach.

For homework, I want you to do two things.  First, 
rewrite the above equation so that it is linear in the 
modelling parameters (e.g., pick some new ones it 
is linear in).  Second, use the normal equations to 
solve for U, t0, and a from the following data:

    t (sec)    b (m)
    0.0020    0.9449
    0.0040    1.9823
    0.0060    2.7881
    0.0080    3.7089
    0.0100    4.4820
    0.0120    5.2732
    0.0140    5.9525
    0.0160    6.6565
    0.0180    7.4292
    0.0200    7.9978

Plot up this data and your fitted curve.