A kicked football leaves the ground at an angle with a velocity of 20.0 m/s as shown in the figure. Suppose the kick is attempted 36.0 m from the goalposts, whose crossbar is 3.05 m above the ground. angle 37
Incomplete.
To find out whether the football reaches the goalposts, we need to analyze the motion of the ball in the x and y directions separately.
Given:
Initial velocity (v₀) = 20.0 m/s
Angle (θ) = 37 degrees
Distance to the goalposts (d) = 36.0 m
Height of the goalposts (h) = 3.05 m
First, let's find the time (t) it takes for the ball to reach the goalposts in the x direction.
Using the formula for horizontal distance:
d = v₀ * cos(θ) * t
Rearranging the formula:
t = d / (v₀ * cos(θ))
Substituting the given values:
t = 36.0 m / (20.0 m/s * cos(37°))
Calculating:
t ≈ 2.26 s
Now, let's find the maximum height (H) the ball reaches in the y direction.
Using the formula for maximum height in projectile motion:
H = (v₀ * sin(θ))² / (2 * g)
where g is the acceleration due to gravity (approximately 9.81 m/s²).
Substituting the given values:
H = (20.0 m/s * sin(37°))² / (2 * 9.81 m/s²)
Calculating:
H ≈ 6.80 m
Finally, let's check if the ball clears the crossbar height. If the maximum height (H) is greater than or equal to the crossbar height (h), the ball clears the crossbar.
Comparing the values:
6.80 m ≥ 3.05 m
Since the maximum height is greater than the crossbar height, the football clears the crossbar successfully.
To determine whether or not the football will clear the goalposts, we need to calculate the maximum height reached by the football and the horizontal distance it travels.
First, let's break down the initial velocity of the football into its horizontal and vertical components.
The vertical component of the velocity can be calculated using the formula:
Vy = V * sin(theta)
where V is the magnitude of the velocity (20.0 m/s) and theta is the angle of the kick (37°).
Vy = 20.0 m/s * sin(37°)
Vy ≈ 12.0 m/s
The horizontal component of the velocity can be calculated using the formula:
Vx = V * cos(theta)
where V is the magnitude of the velocity (20.0 m/s) and theta is the angle of the kick (37°).
Vx = 20.0 m/s * cos(37°)
Vx ≈ 16.0 m/s
Next, we can calculate the time it takes for the football to reach its maximum height.
We know that the vertical motion of the football is subject to acceleration due to gravity (approximately 9.8 m/s^2). The time taken to reach maximum height (t) can be calculated using the formula:
t = Vy / g
where Vy is the vertical component of the velocity (12.0 m/s) and g is the acceleration due to gravity (9.8 m/s^2).
t = 12.0 m/s / 9.8 m/s^2
t ≈ 1.22 s
Now, we can calculate the maximum height reached by the football.
The vertical motion formula for free fall is:
y = yt + Vy * t - 0.5 * g * t^2
where y is the maximum height reached, /yt is the initial vertical height (0 m), Vy is the vertical component of the velocity (12.0 m/s), g is the acceleration due to gravity (9.8 m/s^2), and t is the time taken to reach maximum height (1.22 s).
y = 0 + 12.0 m/s * 1.22 s - 0.5 * 9.8 m/s^2 * (1.22 s)^2
y ≈ 9.39 m
The football reaches a maximum height of approximately 9.39 meters.
Finally, we can calculate the horizontal distance traveled by the football using the formula:
x = Vx * t
where x is the distance traveled, Vx is the horizontal component of the velocity (16.0 m/s), and t is the time taken to reach maximum height (1.22 s).
x = 16.0 m/s * 1.22 s
x ≈ 19.52 m
The football travels a horizontal distance of approximately 19.52 meters.
Now, to determine whether the football clears the goalposts, we need to check if the height reached by the football (9.39 m) is greater than the height of the crossbar (3.05 m).
Since 9.39 m is greater than 3.05 m, the football does clear the goalposts.