Conic Sections: Parabola
Control a virtual water hose to understand parabolic trajectories. Adjust launch angle and initial velocity to hit a moving target.
Help & Instructions
â–¼- **Set Angle:** Adjust the **Launch Angle** (degrees) of the hose.
- **Set Velocity:** Adjust the **Initial Velocity** (m/s) of the water stream.
- **Fire:** Click "Launch Water" to spray the water.
- **Hit the Target:** Aim to hit the yellow moving target. Each hit scores points!
- **Reset:** Click "Reset Game" to start over with a new target.
- Understand the parabolic path of a projectile.
- Recognize how initial velocity and launch angle affect range and maximum height.
- Connect mathematical equations of parabolas to real-world physics.
Projectile Control 💦
Trajectory Info
Max Height: 0.00 m
Range: 0.00 m
The path of a projectile under constant gravity (neglecting air resistance) is a **parabola**. The trajectory can be described by a quadratic equation, where vertical position ($y$) is a function of horizontal position ($x$): $$y = x \tan(\theta) - \frac{g x^2}{2 (v_0 \cos(\theta))^2}$$ Here, $\theta$ is the launch angle, $v_0$ is the initial velocity, and $g$ is the acceleration due to gravity ($9.8 \, m/s^2$).
The Physics Behind the Fun
- **Range (horizontal distance):** $$R = \frac{v_0^2 \sin(2\theta)}{g}$$
- **Max Height:** $$H = \frac{v_0^2 \sin^2(\theta)}{2g}$$
- The maximum range occurs at a launch angle of **45 degrees**.
Understanding parabolas and projectile motion is fundamental in:
- **Sports:** Analyzing throws (basketball, javelin, golf) for optimal trajectories.
- **Military/Defense:** Ballistics of missiles and artillery shells.
- **Engineering:** Designing water fountains, ramps, and roller coasters.
