A Firework Is Launched Into The Air From Ground Level With An Initial Velocity Of 128 Ft/s 128 \, \text{ft/s} 128 Ft/s . If The Acceleration Is F / S 2 F/\text{s}^2 F / S 2 , What Is The Maximum Height Reached By The Firework?Given The Equation $h(t) = At^2 + Vt

Mar 13, 2025 by ADMIN 263 views

A Firework's Ascent: Calculating the Maximum Height Reached

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Introduction

Fireworks are a staple of celebrations and festivities around the world. Their vibrant colors and mesmerizing patterns light up the night sky, creating an unforgettable experience for onlookers. However, have you ever wondered how these pyrotechnic devices manage to reach such great heights? In this article, we will delve into the physics behind a firework's ascent, exploring the factors that contribute to its maximum height.

The Physics of Projectile Motion

When a firework is launched into the air, it follows a curved trajectory under the influence of gravity. This type of motion is known as projectile motion, which is a fundamental concept in physics. The trajectory of a projectile can be described using the equations of motion, which relate the position, velocity, and acceleration of an object over time.

The Equation of Motion

The equation of motion for an object under constant acceleration is given by:

h(t) = at^2 + vt + h_0

where:

$h(t)$ is the height of the object at time $t$
$a$ is the acceleration due to gravity (in this case, $f/\text{s}^2$ )
$v$ is the initial velocity of the firework (in this case, $128 \, \text{ft/s}$ )
$h_0$ is the initial height of the firework (in this case, $0$ ft)

Calculating the Maximum Height

To find the maximum height reached by the firework, we need to find the time at which the velocity of the firework is zero. This is because the velocity of an object is the rate of change of its position with respect to time. When the velocity is zero, the object is at its maximum height.

To find the time at which the velocity is zero, we can differentiate the equation of motion with respect to time:

\frac{dh}{dt} = 2at + v

Setting the velocity to zero, we get:

2at + v = 0

Solving for $t$ , we get:

t = -\frac{v}{2a}

Substituting Values

Now that we have the time at which the velocity is zero, we can substitute the values into the equation of motion to find the maximum height:

h(t) = a\left(-\frac{v}{2a}\right)^2 + v\left(-\frac{v}{2a}\right) + h_0

Simplifying the equation, we get:

h(t) = \frac{v^2}{4a} - \frac{v^2}{2a} + h_0

h(t) = -\frac{v^2}{4a} + h_0

Substituting Values (continued)

Now that we have the equation for the maximum height, we can substitute the values into the equation:

h(t) = -\frac{(128 \, \text{ft/s})^2}{4(f/\text{s}^2)} + 0

Simplifying the equation, we get:

h(t) = -\frac{16384}{4f}

h(t) = -\frac{4096}{f}

Conclusion

In this article, we have explored the physics behind a firework's ascent, calculating the maximum height reached by the firework. We have used the equation of motion to find the time at which the velocity is zero, and then substituted the values into the equation to find the maximum height. The result is a function of the acceleration due to gravity, which is a fundamental concept in physics.

Discussion

The maximum height reached by a firework is a function of the acceleration due to gravity, which is a fundamental concept in physics. The acceleration due to gravity is a constant value that depends on the location on Earth, with a value of $32.2 \, \text{ft/s}^2$ at the surface.

The maximum height reached by a firework is also dependent on the initial velocity of the firework, which is a function of the thrust produced by the firework's engine. The thrust produced by the firework's engine is a function of the amount of fuel burned, which is a function of the time of flight.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Additional Resources

[1] Khan Academy. (n.d.). Projectile Motion. Retrieved from https://www.khanacademy.org/science/physics/projectile-motion
[2] Physics Classroom. (n.d.). Projectile Motion. Retrieved from https://www.physicsclassroom.com/class/vectors/Lesson-1/Projectile-Motion

Note: The values used in this article are for illustrative purposes only and may not reflect real-world values.

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Q: What is the maximum height reached by a firework?

A: The maximum height reached by a firework is a function of the acceleration due to gravity and the initial velocity of the firework. The equation for the maximum height is given by:

h(t) = -\frac{v^2}{4a} + h_0

where:

$h(t)$ is the height of the firework at time $t$
$a$ is the acceleration due to gravity (in this case, $f/\text{s}^2$ )
$v$ is the initial velocity of the firework (in this case, $128 \, \text{ft/s}$ )
$h_0$ is the initial height of the firework (in this case, $0$ ft)

Q: What is the relationship between the acceleration due to gravity and the maximum height reached by a firework?

A: The acceleration due to gravity is a constant value that depends on the location on Earth, with a value of $32.2 \, \text{ft/s}^2$ at the surface. The maximum height reached by a firework is inversely proportional to the acceleration due to gravity. This means that as the acceleration due to gravity increases, the maximum height reached by the firework decreases.

Q: How does the initial velocity of a firework affect its maximum height?

A: The initial velocity of a firework affects its maximum height in a non-linear way. As the initial velocity increases, the maximum height reached by the firework also increases, but at a decreasing rate. This is because the firework's velocity is constantly decreasing due to the acceleration due to gravity, so the firework's height is constantly decreasing as well.

Q: What is the role of the firework's engine in determining its maximum height?

A: The firework's engine plays a crucial role in determining its maximum height. The engine produces a thrust that propels the firework upward, and the amount of thrust produced determines the firework's initial velocity. The initial velocity, in turn, determines the maximum height reached by the firework.

Q: Can a firework reach a maximum height greater than the distance from the ground to the firework's launch site?

A: No, a firework cannot reach a maximum height greater than the distance from the ground to the firework's launch site. This is because the firework's velocity is constantly decreasing due to the acceleration due to gravity, so the firework's height is constantly decreasing as well. At some point, the firework's height will be equal to the distance from the ground to the launch site, and the firework will begin to fall back to the ground.

Q: How does the air resistance affect the maximum height reached by a firework?

A: Air resistance can affect the maximum height reached by a firework, but its effect is typically small compared to the effect of the acceleration due to gravity. Air resistance can cause the firework's velocity to decrease more quickly, which can result in a lower maximum height.

Q: Can a firework reach a maximum height greater than the distance from the ground to the firework's launch site if it is launched at an angle?

A: Yes, a firework can reach a maximum height greater than the distance from the ground to the firework's launch site if it is launched at an angle. This is because the firework's velocity is not constant, but rather changes as the firework moves upward. By launching the firework at an angle, the firework's velocity can be increased, which can result in a higher maximum height.

Q: How does the firework's shape and size affect its maximum height?

A: The firework's shape and size can affect its maximum height, but their effect is typically small compared to the effect of the acceleration due to gravity. A firework with a larger size and a more aerodynamic shape may be able to reach a higher maximum height due to its increased thrust and reduced air resistance.

Q: Can a firework reach a maximum height greater than the distance from the ground to the firework's launch site if it is launched from a high altitude?

A: Yes, a firework can reach a maximum height greater than the distance from the ground to the firework's launch site if it is launched from a high altitude. This is because the firework's initial velocity is greater than if it were launched from the ground, which can result in a higher maximum height.

Q: How does the firework's color and pattern affect its maximum height?

A: The firework's color and pattern do not affect its maximum height. The maximum height reached by a firework is determined by its initial velocity, acceleration due to gravity, and air resistance, not by its color or pattern.

Q: Can a firework reach a maximum height greater than the distance from the ground to the firework's launch site if it is launched in a vacuum?

A: Yes, a firework can reach a maximum height greater than the distance from the ground to the firework's launch site if it is launched in a vacuum. This is because there is no air resistance in a vacuum, which means that the firework's velocity is not affected by air resistance. As a result, the firework's maximum height can be greater than the distance from the ground to the launch site.