Understanding the Domain of a Function in the Context of Projectile Motion

Oct 13, 2025 by recoveryshake.com 74 views

Understanding the Domain of a Function in the Context of Projectile Motion

When a football is thrown into the air, it follows a curved trajectory under the influence of gravity. The maximum height reached by the football and the distance it travels from the point of throw are crucial parameters in understanding the motion. In this article, we will explore the domain of a function in the context of projectile motion, using the given scenario as a case study.

A football is thrown into the air, reaching a maximum height of 18 yards. It then lands 78 yards from where it was thrown. We need to determine the correct domain of the function that describes the position of the football as a function of time.

To analyze the motion of the football, we need to consider the vertical and horizontal components of the motion separately. The vertical component is affected by gravity, while the horizontal component is not. Since the football is thrown upwards, it will eventually reach a maximum height and then fall back down.

The Vertical Component

The maximum height reached by the football is 18 yards. This means that the vertical component of the motion is bounded by the maximum height. In other words, the vertical position of the football is always between 0 and 18 yards.

The Horizontal Component

The distance traveled by the football from the point of throw is 78 yards. This means that the horizontal component of the motion is bounded by the distance traveled. In other words, the horizontal position of the football is always between 0 and 78 yards.

Now that we have analyzed the vertical and horizontal components of the motion, we can determine the correct domain of the function that describes the position of the football as a function of time.

The domain of a function is the set of all possible input values for which the function is defined. In this case, the input value is time, and the output value is the position of the football.

Since the vertical position of the football is always between 0 and 18 yards, and the horizontal position is always between 0 and 78 yards, the correct domain of the function is:

$0 \leq x \leq 78$

This means that the function is defined for all values of time between 0 and 78 yards.

In conclusion, the correct domain of the function that describes the position of the football as a function of time is $0 \leq x \leq 78$ . This is because the vertical position of the football is always between 0 and 18 yards, and the horizontal position is always between 0 and 78 yards.

The domain of a function is a crucial concept in mathematics, and it is essential to understand it in the context of real-world problems. In this article, we have seen how to determine the domain of a function in the context of projectile motion.

The correct domain of the function is $0 \leq x \leq 78$ , which means that the function is defined for all values of time between 0 and 78 yards.

[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.

The domain of a function is a set of all possible input values for which the function is defined. In this case, the input value is time, and the output value is the position of the football. The correct domain of the function is $0 \leq x \leq 78$ , which means that the function is defined for all values of time between 0 and 78 yards.
A Football is Thrown into the Air: A Q&A on Projectile Motion

In our previous article, we explored the domain of a function in the context of projectile motion, using the scenario of a football being thrown into the air as a case study. In this article, we will continue to delve deeper into the world of projectile motion, answering some of the most frequently asked questions related to this topic.

Q: What is projectile motion?

A: Projectile motion is the motion of an object that is thrown or projected into the air, under the influence of gravity. The object follows a curved trajectory, with a vertical component of motion affected by gravity and a horizontal component of motion that is unaffected by gravity.

Q: What are the key factors that affect projectile motion?

A: The key factors that affect projectile motion are:

Initial velocity: The speed and direction at which the object is thrown.
Angle of projection: The angle at which the object is thrown relative to the horizontal.
Gravity: The acceleration due to gravity, which affects the vertical component of motion.
Air resistance: The force that opposes the motion of the object, which can affect the horizontal component of motion.

Q: How do you calculate the maximum height reached by a projectile?

A: To calculate the maximum height reached by a projectile, you can use the following formula:

h = (v0^2 * sin^2(θ)) / (2 * g)

where: h = maximum height v0 = initial velocity θ = angle of projection g = acceleration due to gravity

Q: How do you calculate the range of a projectile?

A: To calculate the range of a projectile, you can use the following formula:

R = (v0^2 * sin(2θ)) / g

where: R = range v0 = initial velocity θ = angle of projection g = acceleration due to gravity

Q: What is the difference between the horizontal and vertical components of motion?

A: The horizontal component of motion is the component of motion that is unaffected by gravity, while the vertical component of motion is the component of motion that is affected by gravity. The horizontal component of motion is constant, while the vertical component of motion is changing due to the acceleration due to gravity.

Q: How do you determine the domain of a function in the context of projectile motion?

A: To determine the domain of a function in the context of projectile motion, you need to consider the vertical and horizontal components of motion separately. The vertical component of motion is bounded by the maximum height, while the horizontal component of motion is bounded by the range.

In conclusion, projectile motion is a complex topic that involves the interaction of several key factors, including initial velocity, angle of projection, gravity, and air resistance. By understanding these factors and how they affect the motion of a projectile, you can calculate the maximum height reached, the range, and the domain of a function in the context of projectile motion.

[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.

Projectile motion is a fundamental concept in physics that has numerous applications in real-world scenarios, including sports, engineering, and astronomy. By understanding the key factors that affect projectile motion and how to calculate the maximum height reached, the range, and the domain of a function, you can gain a deeper appreciation for the beauty and complexity of this topic.