Consider a capapult that fires a projects into the air with an initial velocity
. The free-flying projectile is subjected to air friction and a gravitional force. Given a desired target 
and an initial velocity , find the optimal shooting angle 
of the catapult that minimizes the distance between the target and the trajectory of the fired projectile.
. The free-flying projectile is subjected to air friction and a gravitional force. Given a desired target and an initial velocity , find the optimal shooting angle of the catapult that minimizes the distance between the target and the trajectory of the fired projectile. tip 1: Consider the states 
and 
as the x- and y-position of the projectile, and 
and 
as the x- and y-velocity. Then, the trajectory of the projectile can be found by solving the following ordinary differential equation (ODE):
and as the x- and y-position of the projectile, and and as the x- and y-velocity. Then, the trajectory of the projectile can be found by solving the following ordinary differential equation (ODE):
, 
, 
. where 
, and 
is the friction coefficient between the air and the projectile. Use the ode45.m function to compute the trajectory of the projectile with initial conditions 
. Plotting vs. will result in the x-y trajectory of the projectile, as shown in the figure below.
, and is the friction coefficient between the air and the projectile. Use the ode45.m function to compute the trajectory of the projectile with initial conditions . Plotting vs. will result in the x-y trajectory of the projectile, as shown in the figure below.tip 2: Use the following update law, to incrementally update the shooting angle 
:
:
where 
the smallest Euclidean distance between the trajectory of the projectile and the target 
, 
is a difference angle, and 
an update parameter.
the smallest Euclidean distance between the trajectory of the projectile and the target , is a difference angle, and an update parameter.Example of algorithm's numerical result:
theta = catapult(25,3,25)
theta =
0.8431
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Nice problem. There should be much more problems involving solution of differential equations on Cody. :-)
I have tried many ways of getting isequal( round(...,2),y_correct) to evaluate to 'true' when round(...,2) is, in fact, equal to y_correct. But, they all fail on the Cody server. Why not just test whether abs(catapult(...) - y_correct) <= TOL.