Projectile trajectory

Path of a launched game piece, whether it reaches a target, and the speed needed to hit it.

Inputs

Launch speed

Launch angle above horizontal

Launch height

Horizontal distance to target

Target height

Result

Height at target distance

10.07

Miss (above + / below −): 24.86
Speed needed to hit target: 25.99
Time to target: 0.6223
Entry angle at targetNegative means descending: 8.726
Apex height: 10.21
Range to the floor: 29.13

Trajectory

Path
Target

Data table

Distance (ft)	Height (ft)
0	2
2.549	4.757
5.099	6.952
7.648	8.584
10.2	9.654
12.75	10.16
15.3	10.11
17.84	9.491
20.39	8.311
22.94	6.57
25.49	4.266
28.04	1.4
29.13	1.166 × 10^−14

Target: 12 ft, 8 ft

How it’s calculated

Formula

y(x) = h_0 + x \tan\theta - \frac{g \, x^2}{2 v^2 \cos^2\theta}

t = \frac{x}{v \cos\theta}, \qquad h_\text{apex} = h_0 + \frac{v^2 \sin^2\theta}{2g}

v_\text{needed} = \frac{d}{\cos\theta} \sqrt{\frac{g}{2\,(d \tan\theta - (h_t - h_0))}}

With your numbers (converted to SI units)

y(3.658\,\text{m}) = 0.6096\,\text{m} + 3.658\,\text{m} \tan(50^\circ) - \frac{9.807 \times 3.658^2}{2 \times (9.144\,\text{m/s})^2 \cos^2(50^\circ)} = 3.07\,\text{m}

t = \frac{3.658\,\text{m}}{5.878\,\text{m/s}} = 0.6223\,\text{s}

h_\text{apex} = 0.6096\,\text{m} + \frac{(7.005\,\text{m/s})^2}{2 \times 9.807} = 3.111\,\text{m}

v_\text{needed} = \frac{3.658\,\text{m}}{\cos(50^\circ)} \sqrt{\frac{9.807}{5.06\,\text{m}}} = 7.921\,\text{m/s}

What it means

With no air drag, the game piece follows a parabola. The miss distance is how far above (positive) or below (negative) the target height it passes at the target’s distance. The needed speed is the launch speed that passes exactly through the target at this angle; a steeper entry angle generally makes a shot more forgiving.

Assumptions

No air drag, spin or Magnus lift. Light, large game pieces (foam balls) fall noticeably short of this.
The game piece leaves at the launch speed and angle you enter; measure exit speed rather than relying on flywheel speed.

Learn the concepts behind this on resources.simbotics.org · All formulas