To win the game, a place kicker must kick a football from a point 40 m (43.744 yd) from the goal, and the ball must clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 23 m/s at

an angle of 30.9◦ from the horizontal.
The acceleration of gravity is
9.8 m/s^2.
By how much vertical distance does the ball clear the crossbar?
Answer in units of

To determine how much vertical distance the ball clears the crossbar, we need to analyze the motion of the football.

First, we need to break down the initial velocity of the ball into its vertical and horizontal components. The velocity of 23 m/s at an angle of 30.9 degrees can be separated into:

Vertical component (V_y) = 23 m/s * sin(30.9 degrees)
Horizontal component (V_x) = 23 m/s * cos(30.9 degrees)

To find the time it takes for the ball to reach the crossbar, we can use the equation of motion for the vertical direction:

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

Since the initial vertical displacement (y) is 3.05 m and the acceleration due to gravity (g) is -9.8 m/s^2, we can rearrange the equation to solve for time (t):

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

Substituting the values, we have:

3.05 m = (23 m/s * sin(30.9 degrees)) * t - (1/2) * (9.8 m/s^2) * t^2

Simplifying the equation, we get a quadratic equation:

(1/2) * (9.8 m/s^2) * t^2 - (23 m/s * sin(30.9 degrees)) * t + 3.05 m = 0

Solving this quadratic equation will give us the positive value of time, which represents the time it takes for the ball to reach the crossbar.

Once we have the time, we can substitute it back into the equation for vertical displacement:

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

This will give us the vertical distance the ball clears the crossbar.

Note: Since you have not specified the units for vertical distance, the answer will be in whatever unit the initial vertical displacement (3.05 m) is given in.