To win the game, a place kicker must kick a
football from a point 30 m (32.808 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 19 m/s at
an angle of 38.1
◦
from the horizontal.
The acceleration of gravity is 9.8 m/s
2
.
By how much vertical distance does the ball
clear the crossbar?
To determine the vertical distance by which the ball clears the crossbar, we first need to find the maximum height the ball reaches in its trajectory.
Step 1: Split the initial velocity into its vertical and horizontal components.
The vertical component of the initial velocity, which determines the height reached, can be calculated using the equation:
V_0y = V_0 * sin(Θ)
Given:
V_0 = 19 m/s (initial speed)
Θ = 38.1° (angle of elevation)
V_0y = 19 m/s * sin(38.1°)
V_0y ≈ 11.537 m/s (vertical component of velocity)
Step 2: Calculate the time it takes for the ball to reach its highest point.
The time it takes for the ball to reach its highest point, also known as the "time of flight," can be determined using the equation:
t = V_0y / g
Given:
g = 9.8 m/s^2 (acceleration due to gravity)
V_0y = 11.537 m/s (vertical component of velocity)
t = 11.537 m/s / 9.8 m/s^2
t ≈ 1.178 seconds (time of flight)
Step 3: Calculate the maximum height reached by the ball.
The maximum height reached by the ball, denoted by 'h,' can be calculated using the equation:
h = V_0y^2 / (2 * g)
Given:
V_0y = 11.537 m/s (vertical component of velocity)
g = 9.8 m/s^2 (acceleration due to gravity)
h = (11.537 m/s)^2 / (2 * 9.8 m/s^2)
h ≈ 6.884 meters (maximum height reached)
Step 4: Calculate the vertical distance cleared over the crossbar.
To determine the vertical distance by which the ball clears the crossbar, we subtract the height of the crossbar from the maximum height reached.
Given:
Height of the crossbar = 3.05 meters
Vertical distance cleared = Maximum height reached - Height of the crossbar
Vertical distance cleared = 6.884 meters - 3.05 meters
Vertical distance cleared ≈ 3.834 meters
Therefore, the ball clears the crossbar by approximately 3.834 meters in terms of vertical distance.