A placemaker must kick a football from a point 36 m (about 39yd) 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 velocity of 20.0 m/s at an angle 53 degrees to the horizontal. By how much does the ball clear or fall short of clearing the crossbar?

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To determine how much the ball clears or falls short of clearing the crossbar, we need to find the maximum height the ball reaches during its flight path.

We can start by analyzing the vertical component of the ball's initial velocity. Since the ball was kicked at an angle of 53 degrees to the horizontal, we can find the vertical component by multiplying the initial velocity (20.0 m/s) by the sine of the angle (53 degrees).

Vertical component = initial velocity * sin(angle)
= 20.0 m/s * sin(53 degrees)
≈ 20.0 m/s * 0.800
≈ 16.0 m/s

Now, we can use this vertical component to find the maximum height reached by the ball using the kinematic equation:

Final velocity squared = initial velocity squared + (2 * acceleration * displacement)

At its maximum height, the final velocity of the ball will be zero since it reaches a point where it momentarily stops moving vertically. The initial velocity is the vertical component we just calculated, and the acceleration is the acceleration due to gravity, which is approximately -9.8 m/s^2 (taking downward as negative). The displacement is the maximum height we want to find.

Plugging in the values into the kinematic equation:

0 = (16.0 m/s)^2 + 2 * (-9.8 m/s^2) * displacement

Simplifying the equation gives us:

0 = 256 m^2/s^2 - 19.6 m/s^2 * displacement

Solving for the displacement (maximum height):

displacement = 256 m^2/s^2 / 19.6 m/s^2
≈ 13.06 m

Therefore, the ball reaches a height of approximately 13.06 meters above the ground.

Since the crossbar is located at a height of 3.05 meters, we can find the difference between the maximum height and the height of the crossbar to determine if the ball clears or falls short:

Difference = maximum height - height of crossbar
= 13.06 m - 3.05 m
≈ 10.01 m

Hence, the ball clears the crossbar by approximately 10.01 meters.