Mark saxby is set to kick the field goal to win the state championship for Olathe East. He is 47.0m away from the goalposts and he kicks the ball at a speed of 25.0m/s at an angle of 30.0 degrees.
(a.) If the cross bar on the goalposts is 3.00m above the ground will the ball make it over the cross bar? Justify your answer
(b.) And if it does make it by how much will it clear?
hf=hi+vi'*t-4.8t^2 where is the time in air. You can solve that by
distance horizontal=vh*t
t=47/25cosTheta
Now put that time into the first equaiton, and see if hf is 3m or more.
To determine if the ball will make it over the crossbar and how much it will clear it by, we need to break down the motion of the ball into its horizontal and vertical components.
(a.) First, let's break down the initial velocity of the ball into its horizontal and vertical components. The horizontal component can be found using the formula:
Vx = V * cos(theta)
where V is the initial velocity of the ball (25.0 m/s) and theta is the angle at which it was kicked (30.0 degrees). Substituting the values, we get:
Vx = 25.0 m/s * cos(30.0 degrees)
Vx = 25.0 m/s * 0.866
Vx ≈ 21.65 m/s
The vertical component can be found using the formula:
Vy = V * sin(theta)
Substituting the values, we get:
Vy = 25.0 m/s * sin(30.0 degrees)
Vy = 25.0 m/s * 0.5
Vy = 12.5 m/s
Now that we have the horizontal and vertical components of the initial velocity, we can analyze the motion of the ball using the equations of motion. The time taken, trajectory, and maximum height reached by the ball can be calculated.
To determine the time it takes for the ball to reach the goalposts, we can use the vertical component of velocity (Vy) and the acceleration due to gravity (g) which is approximately 9.8 m/s².
The formula for the time of flight is:
t = (2 * Vy) / g
Substituting the values, we get:
t = (2 * 12.5 m/s) / 9.8 m/s²
t ≈ 2.55 seconds
The horizontal distance covered by the ball can be calculated using the horizontal component of velocity (Vx) and the time of flight (t). The formula for horizontal distance is:
distance = Vx * t
Substituting the values, we get:
distance = 21.65 m/s * 2.55 s
distance ≈ 55.28 meters
Since the distance covered by the ball is greater than the initial distance from the goalposts (47.0 meters), we can conclude that the ball will reach the goalposts.
(b.) To determine how much the ball will clear the crossbar by, we need to find the maximum height reached by the ball and compare it to the height of the crossbar.
The formula for the maximum height (h) reached by a projectile is:
h = (Vy²) / (2 * g)
Substituting the values, we get:
h = (12.5 m/s)² / (2 * 9.8 m/s²)
h ≈ 8.06 meters
Comparing this height to the height of the crossbar (3.00 meters), we can see that the ball will clear the crossbar by approximately:
Clearance = h - (height of the crossbar)
Clearance = 8.06 meters - 3.00 meters
Clearance ≈ 5.06 meters
Therefore, the ball will clear the crossbar by approximately 5.06 meters.
To solve this problem, we can break the initial velocity of the ball into horizontal and vertical components.
(a.) First, let's find the vertical component of the initial velocity. We can use the formula:
Vy = V * sin(θ)
Plugging in the values, we have:
Vy = 25.0 m/s * sin(30.0°)
Vy = 12.5 m/s
Now, let's find the time it takes for the ball to reach its highest point. We can use the equation:
Vy = Vo + g * t
where g is the acceleration due to gravity (approximately 9.8 m/s²) and Vy is the vertical component of the initial velocity (12.5 m/s).
12.5 m/s = 0 + 9.8 m/s² * t
Solving for t, we have:
t = 12.5 m/s / 9.8 m/s²
t ≈ 1.27 s
Next, let's find the maximum height reached by the ball. We can use the formula:
y = Vo * t + 0.5 * a * t²
where y is the height, Vo is the initial vertical velocity, t is the time, and a is the acceleration due to gravity (approximately -9.8 m/s²).
Plugging in the values, we have:
y = 12.5 m/s * 1.27 s + 0.5 * (-9.8 m/s²) * (1.27 s)²
y ≈ 9.98 m
Since the crossbar of the goal is 3.00 m above the ground, and the maximum height reached by the ball is approximately 9.98 m, we can see that the ball will clear the crossbar.
(b.) To find how much the ball will clear the crossbar by, we can subtract the height of the crossbar from the maximum height reached by the ball:
Clearance = Maximum height - Height of crossbar
Clearance = 9.98 m - 3.00 m
Clearance ≈ 6.98 m
Therefore, the ball will clear the crossbar by approximately 6.98 meters.