Egwald Mathematics: Geometry  Trajectory of a Projectiles
by
Elmer G. Wiens
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A projectile is fired from a gun that is mounted on a cliff above the sea. What is the range of the gun? At what angle should the gun be fired to hit a ship within its range?
z_{0} = height of the gun above the xy plane.
v_{0} = muzzle velocity = initial velocity of the projectile.
alpha = the angle between the horizontal (the xy plane) and the muzzle of the gun.
z(t) = the height of the projectile t seconds after being fired.
r(t) = the distance of the projectile from the gun after t seconds.
g = pull of gravity.
Then the parametric equations of motion of (r(t), z(t)) are:
r(t) = v_{0} * cos(alpha) * t
z(t) = 1/2 * g * t^^{2} + v_{0} * sin(alpha) * t + z_{0}



The trajectory of the particle for varying angles alpha.


The trajectory of the particle in varying directions.


Change the angle alpha (in radians, try .75) and click "Submit" to view the trajectories of the particle at the new angle.
