Ballistic Trajectory (2-D) Calculator - Computes the maximum height, range, time to impact, and impact velocity of a ballistic projectile

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Ballistic Trajectory (2-D) Calculator

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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 y_o (initial height) results in a negative value of h (maximum height).

Enter parameters:
`v_o`	=	meters/second
`θ°`	=	degrees
`y_o`	=	meters
	Results:
`R (range)`	=	meters
`h (height)`	=	meters
`T (flight)`	=	seconds
`v_f`	=	meters/second

Apply rounding No rounding

Notes

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·t²

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·t²

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·t²

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_xf² + v_yf²)

In which:

v_xf =  v_xo

v_yf = -g·t_fall
Copyright © 2004, Stephen R. Schmitt