A football is kicked from the ground at an initial angle of 35.0° and an unknown initial speed. A 3.05 m tall goal post is 45.0 m away. With what minimum speed should it be kicked to hit the horizontal crossbar of the goal post?

To find the minimum speed at which the football should be kicked to hit the horizontal crossbar of the goal post, we can use the principles of projectile motion.

Let's break down the problem into horizontal and vertical components of motion. The horizontal motion is constant, and the vertical motion follows the laws of free-fall with constant acceleration due to gravity (g).

Given:
Initial angle (θ) = 35.0°
Height of the goal post (h) = 3.05 m
Distance to the goal post (d) = 45.0 m

First, we need to calculate the time taken for the football to reach the goal post in horizontal motion. For this, we can use the horizontal velocity (Vx) and the distance to calculate time:

Horizontal velocity (Vx) = Initial speed (V₀) * cos(θ)

Next, we can calculate the vertical displacement (Δy) of the football when it reaches the goal post:

Vertical displacement (Δy) = h

Now, let's calculate the time taken to reach the goal post using the vertical motion equation:

Δy = V₀ * sin(θ) * t - (1/2) * g * t²

Substituting the values, we get:

3.05m = V₀ * sin(35°) * t - (1/2) * 9.8 m/s² * t²

Simplifying the equation, we get:

4.9t² - 2.55t + 3.05 = 0

To solve this quadratic equation, we can use the quadratic formula:

t = (-b ± √(b² - 4ac)) / 2a

Where a = 4.9, b = -2.55, and c = 3.05

Solving for t, we get two possible values. We can use the positive value for t since we are interested in the time taken for the ball to reach the goal post:

t = 1.16 s (approx.)

Now, let's find the horizontal velocity:

Vx = V₀ * cos(35°)

Now, we know that the horizontal distance traveled by the ball is the product of the horizontal velocity and the time taken:

Distance (d) = Vx * t

Substituting the known values, we get:

45.0m = V₀ * cos(35°) * 1.16s

Solving for V₀, we find:

V₀ = 45.0m / (cos(35°) * 1.16s)

V₀ ≈ 40.9 m/s

Therefore, the minimum speed at which the football should be kicked to hit the horizontal crossbar of the goal post is approximately 40.9 m/s.