Ballistic Trajectory (2-D) Calculator
by Stephen R. Schmitt
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About
This calculator 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).
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The source code
The motion of an object moving near the surface of the earth can be described using the equations:
| x = xo + vxot | (1) |
| y = yo + vyot − 0.5gt2 | (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:
| vxo = vo cos θ |
| vyo = vo sin 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 | |||
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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 | |||||
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The total flight time of the projectile is then:
| T = trise + tfall |
From this, equation (1) gives the maximum range:
| R = vxotflight |
The projectile speed at impact vf is determined by applying the Pythagorean Theorem:
| vf = | √ | vxf2 + vyf2 |
In which:
| vxf = vxo |
| vyf = −gtfall |