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This calculator (by Stephen R. Schmitt) computes the maximum height, range, time to impact, and impact velocity of a ballistic projectile. Computations are based of the acceleration of gravity on the earth's surface (9.81 m/s/s); atmospheric drag is neglected. The program is operated by entering the initial velocity, initial angle, and height above the surface of the projectile; selecting the rounding option desired, and then pressing the Calculate button. All entries are cleared by pressing the Clear button. If the program returns the error message:   cannot solve  then either: the initial angle is outside the range 0...90, or the velocity is negative, or a negative value for yo (initial height) results in a negative value of h (maximum height).

Ballistic Trajectory (2-D) Calculator
Enter parameters:
vo = meters/second
θ = degrees
yo = meters
R (range) = meters
h (height) = meters
T (flight) = seconds
vf = meters/second

Apply rounding   No rounding


The motion of an object moving near the surface of the earth can be described using the equations:

(1): x = xo + vxot

(2): y = yo + vyot - 0.5gt2

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:

vxo = vocos θ

vyo = vosin θ

Then the calculator computes the time to reach the maximum height by finding the time at which vertical velocity becomes zero:

vy = vyo - gt

trise = vyo/g

Maximum height is obtained by substitution of this time into equation (2).

h = yo + vyot - 0.5gt2

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 - 0.5gt2

tfall = √(2h/g)

The total flight time of the projectile is then:

tflight = trise + tfall 

From this, equation (1) gives the maximum range:

range = vxotflight

The projectile speed at impact vf is determined by applying the Pythagorean Theorem:

vf = √(vxf2 + vyf2)

In which:

vxf =  vxo

vyf = -gtfall
Copyright 2004, Stephen R. Schmitt