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Ballistic Trajectory (2-D) Calculator

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).

 Enter parameters: vo = meters/second θ° = degrees yo = meters Results: R (range) = meters h (height) = meters T (flight) = seconds vf = 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 = xo + vxo·t

(2): y = yo + vyo·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:

```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 - g·t

trise = vyo/g
```

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

```h = yo + vyo·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

tfall = √(2·h/g)
```

The total flight time of the projectile is then:

```tflight = trise + tfall
```

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

```range = vxo·tflight
```

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

```vf = √(vxf2 + vyf2)
```

In which:

```vxf =  vxo

vyf = -g·tfall
```