Project 1
Part 2
The objective of this project is to accurately predict the trajectory of a solid object. In this
part of the project we will attempt to incorporate the air resistance using linear relationship,
y 00 (t) = −g − by 0 ,
x00 (t) = −bx0 ,
where (x(t), y(t)) are the coordinates of the center of mass. The object is launched with initial
velocity
x0 (0) = v0 cos(θ), y 0 (0) = v0 sin(θ)
and the initial position is given by x(0) = 0, y(0) = y 0 .
1) Develop an analytic solution.
2) Calculate b and v0 from the following data obtained by P. Howard. To calculate b use the
following data: it takes 0.95 sec for a dart to fall from the height 4.06 meter (initial velocity is
zero). Next use the following data to compute v0: the time that takes for a dart fired straight up
from a height 0.39 meters to hit ground is 2.13 seconds.
3) Determine analytically an implicit relation for d(θ) (the distance the dart travels given that
it was shot with angle of inclination θ).
4) Write M-file that plots d(θ) versus θ for θ = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85
degrees. Take y0 = 0.18.
5) Compare the data with an experimental data obtained with y 0 = 0.18 and θ = 5, 10, 15, ..., 85
degrees. The data are given in a separate file along with M-file that will plot the distance traveled
in horizontal direction (see first part of the project).
6) Compute the error in the following way. First use the following lines in your matlab code to
obtain four different data sets, d1, d2, d3, d4 from the data.
>>f id = f open(0 ballistic meter.dat0 ,0 r 0 );
>>y = f scanf (f id,0 %f 0 , inf );
>>z = reshape(y, 2, size(y, 1)/2);
>>q = reshape(z(2, :), 4, 17);
>>d1 = q(1, :);
(1)
>>d2 = q(2, :);
>>d3 = q(3, :);
>>d4 = q(4, :);
Use the following relative error estimate
sP
17
c
2
c
2
c
2
c
2
i=1 [(d1(i) − d (i)) + (d2(i) − d (i)) + (d3(i) − d (i)) + (d4(i) − d (i)) ]
,
P17
2
2
2
2
i=1 [(d1(i)) + (d2(i)) + (d3(i)) + (d4(i)) ]
where dc (i) is the distance computed from part 4 for θ(i). Note that the corresponding error in the
first part of the project is 18.28 percent.
1