Parametric plotting of projectile motion
I'm currently stuck on this calc question
On flat terrain, a projectile with an initial velocity of 190 (ft/sec) is fired at a 20 - foot high vertical wall 1000 feet from the tip of the cannon inclined at an angle a with the horizontal.
x[t] measures the horizontal distance in feet of the projectile down range from the cannon at time t .
y[t] measures the height of the projectile in feet at time t .
Gravity acts on the y[t] component of the position but does not act on the x[t] component. Thus x''[t] = 0 and y''[t] = -32 .
The muzzle speed, s , splits into horizontal and vertical components as follows:
x'[0] = s Cos[a]y'[0] = s Sin[a] .
Saying that the cannon is fired at ground level is to say{x[0], y[0]} == 0 .
For approximately what angles of inclination will the projectile hit the wall or soar over the wall?
So far I was able to obtain the x and y functions with respect to parameter t as follows.
x[t_, s_, a_] = s t Cos[a]
y[t_, s_, a_] = -16 t^2 + s t Sin[a]
Using mathematica, i was able to derive a function for the horizontal range in terms of the angle a as follows.
(9025*Cos[a]*Sin[a])/4
But im stuck on how to proceed from here to obtain the range of angles that provide the projectile to either hit the wall or soar over the wall.
Also since we are instructed to use MAthematica, I had to post the above in the form of the coding from MAthematica
Eliminate $t$ from the second equation. You know the distance to the wall, so $$1000=s t \cos a$$ Take $t$ from this equation (will be a function of $a$) and plug this into the second equation, for the height. So now the $y$ coordinate at the wall will be a function of only $a$ (you don't care when the projectile arrives, just that you want to go over).