Projectile Range Symmetry
Demonstrating Galileo's Discovery of Complementary Angles
60°
(45° + α)
30°
(45° - α)
Range for (45° + α):
0.00 m
Range for (45° - α):
0.00 m
Maximum Range (at 45°):
0.00 m
Time of Flight (45° + α):
0.00 s
Time of Flight (45° - α):
0.00 s
Maximum Height (45° + α):
0.00 m
Maximum Height (45° - α):
0.00 m
Galileo's Discovery:
This simulation demonstrates the principle stated by Galileo in Two New Sciences: "for elevations which exceed or fall short of 45° by equal amounts, the ranges are equal."
Mathematical Proof:
The range of a projectile is given by:
\[R = \frac{v_0^2 \sin 2\theta_0}{g}\]
For angles \((45^\circ + \alpha)\) and \((45^\circ - \alpha)\):
- \(2\theta_0\) becomes \((90^\circ + 2\alpha)\) and \((90^\circ - 2\alpha)\) respectively
- \(\sin(90^\circ + 2\alpha) = \cos 2\alpha\)
- \(\sin(90^\circ - 2\alpha) = \cos 2\alpha\)
Therefore, both angles produce the same range:
\[R = \frac{v_0^2 \cos 2\alpha}{g}\]
Key Observations:
- The maximum range occurs at exactly 45° (when \(\sin 2\theta_0 = 1\))
- Angles equally above and below 45° produce identical ranges
- The higher angle stays in the air longer but has a lower horizontal velocity component
- The lower angle has a greater horizontal velocity component but shorter flight time
- The higher angle reaches a greater maximum height than the lower angle
Interactive Tips:
- Adjust the α angle slider to see how different complementary angles produce the same range
- Change the initial speed to see how it affects all parameters
- Try the slow motion mode to better observe the trajectories
- Notice how the maximum range changes with different initial speeds
