Phil Dawson is a professional place kicker for the Cleveland Browns. On average, he kicks the ball at a 41 degree angle with an initial speed of 70 feet per second. For future reference, goal posts are 10 feet high in the NFL.

a) Write parametric equations to model Dawson's kicks

b) Playing in a dome(no outside conditions), what is the maximum length Dawson can successfully make a field goal?

y = v sinθ t - 16t^2

x = v cosθ t

You have θ and v, so plug those in and solve for x when y=10 to get the max goal distance.

or,
y = x tanθ - 16/(v cosθ)^2 x^2

To write the parametric equations for Dawson's kicks, we need to consider the motion of the ball in both the horizontal and vertical directions.

Let's assume that the positive x-axis represents the horizontal direction and the positive y-axis represents the vertical direction.

a) Writing parametric equations for the horizontal and vertical components of the ball's motion:

For the horizontal motion, we assume there is no air resistance. The horizontal component of the ball's speed remains constant throughout the kick. Let's call it Vx.

Vx = 70 ft/s (constant)

For the vertical motion, we need to consider the effect of gravity. The vertical component of the ball's speed changes with time due to gravity. Let's call it Vy.

The initial vertical component of the ball's speed can be found using trigonometry:

Vy(0) = V * sin(θ)

where V is the initial speed of 70 ft/s and θ is the angle of 41 degrees.

Vy(0) = 70 ft/s * sin(41°)

Now, writing the parametric equations:

x(t) = Vx * t

where t represents the time in seconds.

y(t) = Vy(0) * t - (1/2) * g * t^2

where g is the acceleration due to gravity, which is approximately 32.2 ft/s^2.

Now that we have the parametric equations, we can proceed to the next part of the question.

b) To find the maximum length Dawson can successfully make a field goal in a dome, we need to determine the distance at which the ball passes through the goal post without hitting the crossbar, which is at a height of 10 feet.

First, we need to find the time at which the ball reaches the height of the crossbar, which is 10 feet. Substituting y(t) = 10 ft into the vertical equation, we get:

Vy(0) * t - (1/2) * g * t^2 = 10 ft

Simplifying this equation, we obtain a quadratic equation:

(1/2) * g * t^2 - Vy(0) * t + 10 = 0

By solving this quadratic equation for t, we can find the time at which the ball reaches the height of the crossbar.

Once we determine the time, we can substitute it back into the horizontal equation, x(t), to find the distance traveled by the ball (maximum length of a successful field goal) in a dome.