If a kicker is 45 yards from a goal post, which is 10 feet tall, and he kicks the ball with an initial velocity of 75 feet per second, and an angle of 50 degrees above the horizontal, does the ball clear the goal?
To determine whether the ball clears the goal or not, we need to calculate the trajectory of the ball and see if it reaches a height above 10 feet at any point in its path.
To calculate the trajectory, we can use the equations of motion. The horizontal and vertical components of the ball's initial velocity can be found using trigonometry:
Initial vertical velocity (Vy) = initial velocity (V) * sin(angle)
Initial horizontal velocity (Vx) = initial velocity (V) * cos(angle)
Let's substitute the given values:
Initial vertical velocity = 75 ft/s * sin(50°)
Initial horizontal velocity = 75 ft/s * cos(50°)
We can calculate these values using a calculator. The result is:
Vy ≈ 57.81 ft/s
Vx ≈ 45.60 ft/s
The time it takes for the ball to reach its peak height can be calculated using the formula:
Time to peak (t) = Vy / g
Where g is the acceleration due to gravity. On Earth, g is approximately 32.2 ft/s^2.
Using this equation, we can calculate the time to reach the peak:
t = 57.81 ft/s / 32.2 ft/s^2
t ≈ 1.8 seconds
Now, we can find the maximum height (H) reached by the ball using the formula:
H = Vy^2 / (2 * g)
Plugging in the values:
H ≈ (57.81 ft/s)^2 / (2 * 32.2 ft/s^2)
H ≈ 83.14 ft
Since the maximum height reached by the ball is 83.14 feet, which is greater than the height of the goal post (10 feet), we can conclude that the ball does clear the goal.