The motion of an object moving near the surface
of the earth can be described using the equations:
(1): x = x_{o} + v_{xo}·t
(2): y = y_{o} + v_{yo}·t  0.5·g·t^{2}
The calculator solves these two simultaneous equations to
obtain a description of the ballistic trajectory. The horizontal
and vertical components of initial velocity are determined
from:
v_{xo} = v_{o}·cos θ
v_{yo} = v_{o}·sin θ
Then the calculator computes the time to reach the maximum
height by finding the time at which vertical velocity becomes
zero:
v_{y} = v_{yo}  g·t
t_{rise} = v_{yo}/g
Maximum height is obtained by substitution of this time into
equation (2).
h = y_{o} + v_{yo}·t  0.5·g·t^{2}
Next, the time to fall from the maximum height is computed
by solving equation (2) for an object dropped from the maximum
height with zero initial velocity.
0 = h  0.5·g·t^{2}
t_{fall} = √(2·h/g)
The total flight time of the projectile is then:
t_{flight} = t_{rise} + t_{fall}
From this, equation (1) gives the maximum range:
range = v_{xo}·t_{flight}
The projectile speed at impact v_{f} is determined
by applying the Pythagorean Theorem:
v_{f} = √(v_{xf}^{2} + v_{yf}^{2})
In which:
v_{xf} = v_{xo}
v_{yf} = g·t_{fall}
Copyright © 2004, Stephen R. Schmitt
