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?
To determine how much the ball clears or falls short of clearing the crossbar, we can break down the projectile motion into its horizontal and vertical components.
First, let's find the horizontal distance traveled by the ball. The horizontal component of the initial velocity can be calculated using the equation:
Vx = V * cos(θ)
where Vx is the horizontal component of the velocity, V is the initial velocity (20.0 m/s), and θ is the launch angle (53 degrees).
Vx = 20.0 m/s * cos(53 degrees)
Vx ≈ 20.0 m/s * 0.6018
Vx ≈ 12.04 m/s
Next, we can find the time of flight by using the equation:
t = 2 * Vy / g
where Vy is the vertical component of the velocity and g is the acceleration due to gravity (approximately 9.8 m/s²).
Vy = V * sin(θ)
Vy = 20.0 m/s * sin(53 degrees)
Vy ≈ 20.0 m/s * 0.7986
Vy ≈ 15.97 m/s
t = 2 * 15.97 m/s / 9.8 m/s²
t ≈ 3.27 s
Now that we know the time of flight, we can calculate the maximum height reached by the ball using the equation:
h = Vy² / (2 * g)
h = (15.97 m/s)² / (2 * 9.8 m/s²)
h ≈ 12.85 m
Finally, let's determine the vertical distance covered by the ball between the launch point and the crossbar (39yd - 36m = 9.144m). The ball needs to clear the crossbar by (12.85 m + 3.05 m) - 9.144 m:
Clearance = (12.85 m + 3.05 m) - 9.144 m
Clearance ≈ 6.756 m
Therefore, the ball clears the crossbar by approximately 6.756 meters.