Firework's Maximum Height: Calculation Guide

Hey guys! Ever wondered how high a firework goes before it bursts into a dazzling display? It's a classic physics problem, and we're going to break it down today. We'll use a bit of math, but don't worry, we'll keep it super clear and easy to follow. We're tackling the question: How to calculate the maximum height reached by a firework launched with an initial velocity of 128 ft/s, considering gravity's acceleration?

Understanding the Physics Behind the Flight

Before we dive into the calculations, let's get a grasp on the physics involved. When a firework launches, it's propelled upwards by an initial force, giving it an initial velocity. In our case, that's 128 feet per second (ft/s). However, once it's airborne, gravity starts pulling it back down. This gravitational pull causes the firework to slow down as it ascends. The acceleration due to gravity is approximately -16 ft/s², the negative sign indicates that the acceleration is in the opposite direction to the initial velocity, which is downwards.

So, the firework's velocity decreases until it momentarily reaches zero at its highest point. This is the maximum height we're trying to find. After reaching this peak, gravity takes over, and the firework begins to fall back to earth, accelerating as it descends. The key concept here is that at the maximum height, the firework's instantaneous vertical velocity is zero. This is a crucial piece of information for our calculation.

Key Concepts to Remember

• Initial Velocity: The upward speed at which the firework is launched (128 ft/s in our case).
• Gravity: The constant downward acceleration acting on the firework (-16 ft/s²).
• Maximum Height: The point where the firework's upward velocity momentarily becomes zero.

Knowing these concepts helps us understand the firework's trajectory and allows us to use the appropriate formulas to calculate its maximum height. Now that we have a solid understanding of the physics involved, let's move on to the mathematical side of things. We'll explore the equation that governs the firework's motion and see how we can use it to find the answer.

The Equation of Motion: Our Key to Solving the Puzzle

To figure out the maximum height, we'll use a fundamental equation of motion from physics. This equation describes the height of an object (h) at any given time (t) when it's under constant acceleration. The equation is:

h(t) = at² + v₀t + h₀

Let's break down what each part of this equation means:

• h(t): This represents the height of the firework at time t. It's what we're trying to find, specifically the maximum height.
• a: This is the acceleration due to gravity, which we know is -16 ft/s².
• t: This represents the time elapsed since the firework was launched. We'll need to figure out the time it takes to reach the maximum height.
• v₀: This is the initial velocity of the firework, which is given as 128 ft/s.
• h₀: This is the initial height of the firework. Since it's launched from ground level, h₀ is 0.

Now, let's plug in the values we know into the equation:

h(t) = -16t² + 128t + 0

h(t) = -16t² + 128t

This equation tells us the height of the firework at any time t. But how do we find the maximum height? Remember, at the maximum height, the firework's velocity is momentarily zero. We need to use another equation of motion to find the time at which this happens.

Finding the Time to Reach Maximum Height

The equation that relates final velocity (v), initial velocity (v₀), acceleration (a), and time (t) is:

v = v₀ + a*t

At the maximum height, the final velocity (v) is 0. We know the initial velocity (v₀) is 128 ft/s, and the acceleration (a) is -16 ft/s². Let's plug these values in and solve for t:

0 = 128 + (-16)*t

16t = 128

t = 128 / 16

t = 8 seconds

So, it takes 8 seconds for the firework to reach its maximum height. Now that we know the time, we can plug it back into our height equation to find the maximum height itself.

Calculating the Grand Finale: The Maximum Height

We've found that it takes 8 seconds for the firework to reach its peak. Now, we'll use this time in our height equation to calculate the maximum height. Remember our equation:

h(t) = -16t² + 128t

We'll substitute t with 8 seconds:

h(8) = -16(8)² + 128(8)

h(8) = -16(64) + 1024

h(8) = -1024 + 1024

h(8) = 512 feet

Boom! The maximum height reached by the firework is 512 feet. That's quite a climb!

So, the final answer to our question, How to calculate the maximum height reached by a firework launched with an initial velocity of 128 ft/s, considering gravity's acceleration? is 512 feet. This means the correct option among the choices provided would be C. 512 ft.

Let's recap the steps we took to solve this problem:

1. We understood the physics of the problem, recognizing the effects of initial velocity and gravity.
2. We identified the relevant equation of motion: h(t) = at² + v₀t + h₀.
3. We used another equation of motion, v = v₀ + a*t, to find the time it takes to reach maximum height.
4. We plugged the time back into the height equation to calculate the maximum height.

This problem showcases how physics and math work together to describe the world around us. By understanding the principles of motion and using the right equations, we can predict the trajectory of objects, even something as exciting as a firework!

Real-World Applications and Further Exploration

Understanding projectile motion, like the firework's trajectory, isn't just about solving textbook problems. It has numerous real-world applications! Think about sports – the path of a baseball, a basketball, or a soccer ball all follow the same principles. Engineers use these concepts to design everything from bridges to rockets. Even in fields like forensics, understanding projectile motion can help reconstruct events.

Beyond Fireworks: Exploring Projectile Motion

If you're curious to delve deeper into this topic, here are a few avenues to explore:

• Air Resistance: In our calculation, we ignored air resistance. In reality, air resistance plays a significant role, especially at higher speeds. How would air resistance affect the maximum height of the firework?
• Launch Angle: We assumed the firework was launched straight upwards. What if it was launched at an angle? How would the angle affect the maximum height and the horizontal distance it travels?
• Different Gravitational Environments: How would the maximum height change if the firework was launched on the Moon, where the gravity is much weaker?

These questions can lead to some fascinating investigations. You can even try simulating projectile motion using online tools or programming languages. Experimenting with different parameters and seeing how they affect the trajectory can be a great way to solidify your understanding.

So guys, the next time you see a firework display, you'll not only enjoy the dazzling colors and patterns but also appreciate the physics that makes it all possible. You'll have a deeper understanding of the forces at play and the calculations involved in predicting its flight. Keep exploring, keep questioning, and keep learning! Physics is all around us, making the world a fascinating place to discover. And remember, understanding these principles can help you not only solve problems but also appreciate the beauty and complexity of the world we live in.