To win the game, a place kicker must kick a football from a point 19 m 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 16 m/s at an angle of 59.8 from the horizontal. The acceleration of gravity is 9.8 m/s. By how much vertical distance does the ball clear the crossbar?

To determine the vertical distance by which the ball clears the crossbar, we need to calculate the maximum height the ball reaches and then subtract the height of the crossbar.

To find the maximum height, we can use the vertical motion equation:

h = (v^2 * sin^2θ) / (2g)

where:
h = maximum height
v = initial velocity (16 m/s)
θ = launch angle (59.8 degrees)
g = acceleration due to gravity (9.8 m/s^2)

First, convert the launch angle from degrees to radians:
θ_rad = θ * π / 180

Substituting the given values, we have:
h = (16^2 * sin^2(59.8)) / (2 * 9.8)

Next, we solve for h:
h = (256 * sin^2(59.8)) / 19.6

Now, we can calculate the vertical distance by subtracting the height of the crossbar from the maximum height:

vertical distance = h - height of crossbar = h - 3.05

Finally, substitute the value of h into the equation to calculate the vertical distance by which the ball clears the crossbar.