You are trying out for the position of place-kicker on a professional football team. With the ball teed 50.0 m from the goalposts with a crossbar 3.05 m off the ground, you kick the ball at 25 m/s and 36° above the horizontal.
To determine whether or not you successfully make the kick, we need to calculate the horizontal and vertical components of the ball's velocity, as well as the time it takes for the ball to reach the goalposts.
First, we can calculate the horizontal component of the ball's velocity using the given speed and launch angle:
horizontal velocity = initial speed * cos(angle)
horizontal velocity = 25 m/s * cos(36°)
Next, we can calculate the vertical component of the ball's velocity:
vertical velocity = initial speed * sin(angle)
vertical velocity = 25 m/s * sin(36°)
Now, we can calculate the time it takes for the ball to reach the goalposts. Since we are given the horizontal distance from the teed ball to the goalposts, we can use the horizontal velocity to find the time:
horizontal distance = horizontal velocity * time
time = horizontal distance / horizontal velocity
Using the given distance of 50.0 m:
time = 50.0 m / horizontal velocity
Finally, to determine if the ball clears the crossbar, we need to calculate the vertical position of the ball at that time. We can use the vertical velocity and the time calculated above in the equation:
vertical position = initial vertical velocity * time + (1/2) * acceleration * time^2
where the initial vertical velocity is equal to the vertical velocity, and the acceleration is due to gravity (approximately -9.8 m/s^2).
If the vertical position is greater than the height of the crossbar (3.05 m), then you have successfully made the kick.
However, if the vertical position is lower than the crossbar at that time, then you have not successfully made the kick.
Remember to double-check your calculations and make sure all units are consistent throughout the calculations.