A football player kicks a ball with a speed of 22.7 m/s at an angle of 47° above the horizontal
from a distance of 36 m from the goal line. (a) By how much does the ball clear or fall short of
clearing the crossbar of the goalpost if that bar is 3.05 m high? (b) What is the vertical velocity
of the ball at the time it reaches the goalpost?
To solve this problem, we can break it down into two parts:
(a) To find the vertical distance by which the ball clears or falls short of the crossbar, we need to calculate the maximum height the ball reaches and then determine the difference between that height and the height of the crossbar.
First, let's find the maximum height using the vertical component of the initial velocity and the acceleration due to gravity. The equation for the maximum height (h) is given by:
h = (v^2 * sin^2θ) / (2 * g)
where v is the initial velocity (22.7 m/s), θ is the angle of projection (47°), and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values, we get:
h = (22.7^2 * sin^2(47°)) / (2 * 9.8)
Using a calculator, we find:
h ≈ 11.19 m
Next, we need to determine the difference between the maximum height and the height of the crossbar (3.05 m). So, the ball clears the crossbar by:
11.19 m - 3.05 m ≈ 8.14 m
Therefore, the ball clears the crossbar by approximately 8.14 meters.
(b) To find the vertical velocity of the ball when it reaches the goalpost, we can use the equation for the vertical component of the velocity at any point in time:
v_y = v * sin(θ) - g * t
where v is the initial velocity (22.7 m/s), θ is the angle of projection (47°), g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time it takes for the ball to reach the goalpost.
We know that the horizontal distance traveled by the ball is 36 m. We can find the time taken using the horizontal component of the velocity and the distance:
d = v * cos(θ) * t
Rearranging the equation, we get:
t = d / (v * cos(θ))
Plugging in the values, we have:
t = 36 m / (22.7 m/s * cos(47°))
Using a calculator, we find:
t ≈ 2.083 s
Now, let's substitute the values of v, θ, g, and t back into the formula for vertical velocity:
v_y = 22.7 m/s * sin(47°) - 9.8 m/s^2 * 2.083 s
Calculating this, we get:
v_y ≈ 13.40 m/s
Therefore, the vertical velocity of the ball when it reaches the goalpost is approximately 13.40 m/s.