To win the game, a place kicker must kick a
football from a point 17 m (18.5912 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 15 m/s at
an angle of 58.9
◦
from the horizontal.
The acceleration of gravity is 9.8 m/s
2
.
By how much vertical distance does the ball
clear the crossbar?
Answer in units of m
The horizontal compoent of the football's velocity is and remains, 15.0 cos58.9 = 7.748 m/s
The time it takes the ball to reach the goal post is therefore
t = 17.0/7.748 = 2.19 seconds
Now use that time t to calculate the distance of the ball about the ground at that time, y.
y = 15.0 sin58.9*t -4.9 t^2
= 12.84 t - 4.9 t^2 = ___ m
Is the answer more or less than 3.0 meters? Calculate it and see for yourself.
Why did the football bring a ladder to the game? Because it wanted to clear the crossbar, of course!
To find out how much vertical distance the ball clears the crossbar, we need to break down the initial velocity into its vertical and horizontal components.
The vertical component of the initial velocity can be found using the formula:
Vy = V * sin(θ)
where V is the initial velocity (15 m/s) and θ is the angle (58.9°). Plugging in the values, we have:
Vy = 15 * sin(58.9°)
Now, we can use the formula for projectile motion to find the vertical distance the ball clears the crossbar.
The formula is:
Δy = Vyi * t + (1/2) * a * t^2
where Δy is the vertical distance covered, Vyi is the initial vertical velocity, t is the time of flight, and a is the acceleration due to gravity (9.8 m/s^2).
Since the ball leaves the ground and lands at the same height, we can ignore the first term in the equation.
Simplifying, we have:
Δy = (1/2) * a * t^2
Plugging in the values, we have:
Δy = (1/2) * 9.8 * t^2
To find time of flight (t), we can use the equation:
t = 2 * Vyi / a
Plugging in the values, we have:
t = 2 * Vy / a
= 2 * (15 * sin(58.9°)) / 9.8
Now we can find the vertical distance the ball clears the crossbar by substituting the value of t into the equation for Δy:
Δy = (1/2) * 9.8 * [(2 * (15 * sin(58.9°)) / 9.8)]^2
Calculating this out, we have:
Δy ≈ 5.25 meters
So, the ball clears the crossbar by approximately 5.25 meters. Go, ball, go!
To find the vertical distance by which the ball clears the crossbar, we need to analyze the projectile motion of the ball.
Step 1: Break down the initial velocity into its vertical and horizontal components.
The initial velocity of the ball can be broken down into its vertical and horizontal components using trigonometry.
Given:
Initial speed (v) = 15 m/s
Launch angle (θ) = 58.9°
Vertical component: v_y = v * sin(θ)
v_y = 15 * sin(58.9°)
Horizontal component: v_x = v * cos(θ)
v_x = 15 * cos(58.9°)
Step 2: Calculate the time it takes for the ball to reach its highest point.
The time taken to reach the maximum height (t_max) can be calculated using the formula:
t_max = v_y / g
where g is the acceleration due to gravity (9.8 m/s^2).
Substituting the given values:
t_max = v_y / g
Step 3: Calculate the maximum height reached by the ball.
The maximum height (h_max) reached by the ball can be calculated using the formula:
h_max = (v_y^2) / (2 * g)
where g is the acceleration due to gravity (9.8 m/s^2).
Substituting the given values:
h_max = (v_y^2) / (2 * g)
Step 4: Calculate the total vertical distance by which the ball clears the crossbar.
The total vertical distance (d_clear) by which the ball clears the crossbar is equal to the sum of the maximum height reached and the difference between the maximum height and the height of the crossbar.
Given:
Height of the crossbar = 3.05 m
Substituting the given values:
d_clear = h_max + (h_max - height of crossbar)
Step 5: Determine the numerical value of the total vertical distance by which the ball clears the crossbar.
Substitute the calculated values from previous steps into the formula for d_clear and evaluate:
d_clear = h_max + (h_max - height of crossbar)
To find out how much vertical distance the ball clears the crossbar, we need to calculate the maximum height reached by the ball first.
Using the given information, we can break down the initial velocity of the ball into its vertical and horizontal components.
The vertical component of the initial velocity can be calculated using the formula:
V𝑦 = V * sin(θ)
where V is the initial velocity (15 m/s) and θ is the launch angle (58.9°).
V𝑦 = 15 * sin(58.9°)
V𝑦 ≈ 12.674 m/s
Now, let's determine the time it takes for the ball to reach its maximum height. We can use the formula:
V𝑦 = 𝑔 * 𝑡
where V𝑦 is the vertical component of the initial velocity, 𝑔 is the acceleration due to gravity (9.8 m/s²), and 𝑡 is the time.
12.674 = 9.8 * 𝑡
𝑡 ≈ 1.293 seconds
Next, we can use the formula for displacement in the vertical direction to find the maximum height (h) reached by the ball:
𝑦 = 𝑦0 + 𝑉0𝑦 * 𝑡 - (1/2) * 𝑔 * 𝑡²
Since the ball starts at ground level (𝑦0 = 0) and we're interested in the maximum height, we have:
h = 0 + 12.674 * 1.293 - (1/2) * 9.8 * (1.293)²
h ≈ 10.231 m
So, the ball reaches a maximum height of approximately 10.231 meters.
To find out how much vertical distance the ball clears the crossbar, we need to subtract the height of the crossbar (3.05 m) from the maximum height.
Vertical distance cleared = Maximum height - Height of crossbar
Vertical distance cleared = 10.231 - 3.05
Vertical distance cleared ≈ 7.181 meters
Therefore, the ball clears the crossbar by approximately 7.181 meters.