A Football Is Punted From A Height Of 2.5 Feet Above The Ground With An Initial Vertical Velocity Of 45 Feet Per Second. Which Vertical Motion Model Could Be Used To Find The Height, \[$ H \$\] (in Feet), Of The Football As A Function Of Time,

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Introduction

When a football is punted, it follows a curved trajectory under the influence of gravity. Understanding the motion of the football is crucial in various aspects of the game, including strategy and player safety. In this article, we will explore the vertical motion model that can be used to find the height of the football as a function of time.

The Vertical Motion Model

The vertical motion of the football can be modeled using the equation of motion under constant acceleration. Since the football is subject to the acceleration due to gravity (g), which is approximately 32 feet per second squared, we can use the following equation:

h(t) = h0 + v0t - (1/2)gt^2

where:

  • h(t) is the height of the football at time t
  • h0 is the initial height of the football (2.5 feet in this case)
  • v0 is the initial vertical velocity of the football (45 feet per second in this case)
  • g is the acceleration due to gravity (32 feet per second squared)
  • t is the time in seconds

Understanding the Equation

Let's break down the equation and understand what each term represents.

  • The first term, h0, represents the initial height of the football. This is the height at which the football is punted.
  • The second term, v0t, represents the initial vertical velocity of the football. This is the velocity at which the football is punted.
  • The third term, -(1/2)gt^2, represents the acceleration due to gravity. This term is negative because the acceleration due to gravity is downward, which means it decreases the height of the football.

Applying the Equation

Now that we have the equation, let's apply it to the given problem. We want to find the height of the football as a function of time.

h(t) = 2.5 + 45t - (1/2)(32)t^2

Simplifying the Equation

We can simplify the equation by combining like terms.

h(t) = 2.5 + 45t - 16t^2

Finding the Maximum Height

To find the maximum height of the football, we need to find the time at which the height is maximum. We can do this by taking the derivative of the equation with respect to time and setting it equal to zero.

dh/dt = 45 - 32t = 0

Solving for t, we get:

t = 45/32 = 1.40625 seconds

Finding the Maximum Height

Now that we have the time at which the height is maximum, we can substitute this value into the equation to find the maximum height.

h(1.40625) = 2.5 + 45(1.40625) - 16(1.40625)^2

h(1.40625) = 2.5 + 63.28125 - 31.25781

h(1.40625) = 34.23 feet

Conclusion

In this article, we have explored the vertical motion model that can be used to find the height of a football as a function of time. We have applied the equation of motion under constant acceleration to the given problem and found the maximum height of the football. This model can be used to understand the motion of the football and make predictions about its trajectory.

Discussion

The vertical motion model is a fundamental concept in physics that can be applied to various situations, including projectile motion. The equation of motion under constant acceleration is a powerful tool that can be used to model and analyze complex motion.

References

  • Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
  • Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.

Additional Resources

  • Khan Academy: Projectile Motion
  • MIT OpenCourseWare: Physics 8.01: Classical Mechanics
  • Wolfram Alpha: Projectile Motion Calculator