Introduction:

As a non-mathematician relying on GPT-4 for guidance, I invite experts in the field to validate or debunk this answer. Quadratic equations have a knack for baffling even the most seasoned problem solvers. This post dives into a common issue that people tend to solve incorrectly, and reveals how using a quadratic equation is the best way to tackle it. So, let’s unravel this mathematical enigma together!

# The Problem: Predicting Projectile Motion

Picture a cannonball being launched from a cannon. The goal is to determine the maximum height the cannonball will reach and the horizontal distance it will travel before hitting the ground. This problem may seem simple at first glance, but it’s easy to make mistakes when attempting to solve it.

## The Missteps: Misinterpreting the Scenario

The most common mistake when solving this problem is misinterpreting the scenario as a simple linear equation. A linear equation may help you find the average speed, but it won’t accurately predict the projectile’s motion.

### The Solution: Embracing Quadratic Equations

To solve this problem correctly, we need to rely on the power of quadratic equations. Quadratic equations describe parabolic motion, which perfectly aligns with the trajectory of a projectile, such as a cannonball.

The quadratic formula, `(-b ± √(b²-4ac)) / 2a`, can be used to find the roots (or solutions) of a quadratic equation. In the case of projectile motion, we’ll need two separate quadratic equations: one for the horizontal (x) motion and one for the vertical (y) motion.

##### Breaking Down the Equations

To predict the projectile’s motion, we’ll need to consider the initial velocity, angle of launch, and acceleration due to gravity. Using these factors, we can create our quadratic equations:

1. Horizontal motion: x(t) = v₀x * t
2. Vertical motion: y(t) = v₀y * t – 0.5 * g * t²

Here, `v₀x` and `v₀y` represent the initial horizontal and vertical velocities, `g` is the acceleration due to gravity, and `t` is time.

###### Solving for Maximum Height and Distance

Using the equations above, we can find the maximum height and horizontal distance of the projectile. To do this, we’ll need to find the vertex of the parabolic path, which represents the highest point of the projectile’s trajectory. The time it takes to reach the maximum height can be found by setting the vertical velocity to zero and solving for `t`. With this information, we can then calculate the maximum height and horizontal distance the projectile will travel before hitting the ground.