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
To determine how much vertical distance the ball clears the crossbar, we need to analyze the motion of the ball.
Let's break down the initial velocity of the ball into its vertical and horizontal components. The vertical component of the initial velocity can be found by multiplying the initial velocity (15 m/s) by the sine of the launch angle (58.9°):
Vertical component of initial velocity (Vy) = 15 m/s * sin(58.9°)
The horizontal component of the initial velocity can be found by multiplying the initial velocity (15 m/s) by the cosine of the launch angle (58.9°):
Horizontal component of initial velocity (Vx) = 15 m/s * cos(58.9°)
Now, let's find the time it takes for the ball to reach its maximum height. At the maximum height, the vertical component of the velocity is zero. We can use the kinematic equation:
final vertical velocity (Vfy) = initial vertical velocity (Vy) + acceleration * time
Since Vfy = 0 (at the maximum height) and acceleration = -9.8 m/s^2 (due to gravity acting downward), we can rearrange the equation to solve for time:
time = -Vy / acceleration
Now that we have the time it takes for the ball to reach its maximum height, we can determine the height it reaches. We'll use the kinematic equation:
final vertical position (y) = initial vertical position (0) + initial vertical velocity (Vy) * time + (1/2) * acceleration * time^2
The final vertical position represents the height of the ball when it reaches the maximum height. So, the ball clears the crossbar by the height it reaches minus the height of the crossbar.
To summarize the steps:
1. Find the vertical component of the initial velocity using Vy = 15 m/s * sin(58.9°).
2. Find the time it takes for the ball to reach its maximum height using time = -Vy / acceleration.
3. Calculate the height the ball reaches using final vertical position (y) = 0 + Vy * time + (1/2) * acceleration * time^2.
4. Subtract the height of the crossbar (3.05 m) from the height the ball reaches.
Finally, plug in the values and calculate the result to find the vertical distance the ball clears the crossbar.