Quadratic functions can be used to find the best height for launching a rocket. This is especially true when looking at how projectiles move in the air.

When a rocket is launched, its height, noted as $h(t)$, changes over time $t$. We can describe this change with a quadratic equation like this:

$$h(t) = -at^2 + bt + c$$

Here's what the letters mean:

• $a$ is a positive number that relates to gravity. It's usually about $9.81 \, \text{m/s}^2$.
• $b$ is how fast the rocket is going when it starts its flight (initial velocity).
• $c$ is how high the rocket starts from (initial height).

Key Points:

1. Curved Path: The rocket travels in a curved path called a parabola.

   • The highest point (or peak) of this curve can be found using the formula: $t = -\frac{b}{2a}.$ This tells us the time when the rocket reaches its highest point.

2. Finding Maximum Height:

   • After we find the best time to reach the peak, we can put that time back into the height equation to find out how high the rocket will go.

3. Example Calculation:

   • Let’s say we launch a rocket with a speed of $50 \, \text{m/s}$ starting from the ground. The equation might look like this: $h(t) = -4.905t^2 + 50t.$
   • To find the time it takes to reach the maximum height, we can use the formula: $t = -\frac{50}{2(-4.905)} \approx 5.1 \, \text{s}.$
   • After that, we can plug this time back into our height equation to find out the peak height.

Conclusion:

In simple terms, quadratic equations play a key role in studying how things like rockets move. They help scientists and engineers understand the best way to launch rockets and predict how high they can go.