To win the game, a place kicker must kick a

football from a point 40 m (43.744 yd) 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 23 m/s at
an angle of 30.9

from the horizontal.
The acceleration of gravity is 9.8 m/s
2
.
By how much vertical distance does the ball
clear the crossbar?

Well, if the place kicker wants to clear the crossbar, they better hope they don't kick the ball like I do. I tend to send it straight into the parking lot! In any case, let's try to help your poor kicker out.

First, we need to break down the initial velocity into its horizontal and vertical components. The horizontal component is given by Vx = V * cos(theta), and the vertical component is Vy = V * sin(theta), where V is the initial velocity and theta is the launch angle.

Plugging in the values, we get Vx = 23 m/s * cos(30.9°) and Vy = 23 m/s * sin(30.9°). Solving them gives us Vx = 19.864 m/s and Vy = 11.785 m/s.

Now, let's focus on the vertical motion of the ball. We need to find out how high the ball goes in order to determine if it clears the crossbar. We can use the kinematic equation:

h = (Vy^2) / (2 * g)

Where h is the maximum height reached, Vy is the initial vertical velocity, and g is the acceleration due to gravity.

Plugging in the values, we have h = (11.785 m/s)^2 / (2 * 9.8 m/s^2). Solving this gives us h ≈ 7.03 m.

Since the crossbar is 3.05 m high, we subtract that from the maximum height to find the vertical distance the ball clears the crossbar: 7.03 m - 3.05 m = 3.98 m.

So, the ball clears the crossbar by approximately 3.98 meters. I hope this helps your kicker win the game! Just make sure they don't aim for the parking lot like I would.

To find out how much vertical distance the ball clears the crossbar, we can break down the motion into horizontal and vertical components.

Step 1: Find the vertical component of the initial velocity.
The initial velocity of the ball can be broken down into its vertical and horizontal components using trigonometry. We can use the formula:
Vy = V * sin(theta)
where Vy is the vertical component of the initial velocity, V is the magnitude of the velocity (given as 23 m/s), and theta is the angle with the horizontal (given as 30.9 degrees).

Using this formula:
Vy = 23 m/s * sin(30.9 degrees)
Vy ≈ 11.82 m/s

Step 2: Calculate the time it takes for the ball to reach its maximum height.
The time it takes for the ball to reach its maximum height can be calculated using the formula:
t = Vy / g
where g is the acceleration due to gravity (given as 9.8 m/s^2).

Using this formula:
t = 11.82 m/s / 9.8 m/s^2
t ≈ 1.205 seconds

Step 3: Calculate the vertical distance the ball reaches.
The vertical distance the ball reaches at its maximum height can be calculated using the formula:
d = (Vy^2) / (2 * g)

Using this formula:
d = (11.82 m/s)^2 / (2 * 9.8 m/s^2)
d ≈ 7.089 meters

Step 4: Calculate the total vertical distance the ball clears the crossbar.
The total vertical distance the ball clears the crossbar can be calculated by subtracting the height of the crossbar (3.05 meters) from the maximum height reached by the ball.

Using this formula:
Total vertical distance cleared = Maximum height reached - Height of crossbar
Total vertical distance cleared = 7.089 meters - 3.05 meters
Total vertical distance cleared ≈ 4.039 meters

Therefore, the ball clears the crossbar by approximately 4.039 meters in the vertical direction.

To find the vertical distance by which the ball clears the crossbar, we need to solve for the maximum height reached by the ball.

To do this, we need to break down the initial velocity of the ball into its horizontal and vertical components. The vertical component will help us determine the maximum height reached.

Given:
Initial speed (v) = 23 m/s
Launch angle (θ) = 30.9°
Acceleration due to gravity (g) = 9.8 m/s²

First, we need to find the vertical component of the initial velocity (v_y) using the equation:

v_y = v * sin(θ)

v_y = 23 * sin(30.9°)
v_y = 11.94 m/s

Now, we can use the time of flight (t) formula to find the time it takes for the ball to reach its maximum height:

t = 2 * v_y / g

t = 2 * 11.94 m/s / 9.8 m/s²
t = 2.435 seconds

Next, we can find the maximum height (h) reached by the ball using the formula:

h = v_y * t - (1/2) * g * t²

h = 11.94 m/s * 2.435 s - (1/2) * 9.8 m/s² * (2.435 s)²
h ≈ 14.55 meters

Therefore, the ball clears the crossbar by approximately 14.55 meters vertically.