A place kicker must kick a football from a point 31.2 m from a goal. As a result of the kick, the ball must clear the crossbar, which is 3.05 m high. When kicked the ball leaves the ground with a speed of 21.3 m/s at an angle of 53° to the horizontal. By how much does the ball clear or fall short of clearing the crossbar?
To determine how much the ball clears or falls short of clearing the crossbar, we can break down the motion of the ball into horizontal and vertical components.
First, let's find the time it takes for the ball to reach the crossbar. We can use the vertical component of the motion because that's what determines the height of the ball. We'll use the kinematic equation:
y = y0 + v0y * t - (1/2) * g * t^2
Where:
y = vertical displacement (height of the crossbar) = 3.05 m
y0 = initial vertical position (initial height of the ball) = 0 m
v0y = initial vertical velocity = v0 * sin(θ)
g = acceleration due to gravity = 9.8 m/s^2
t = time
Substituting the known values:
3.05 = 0 + (21.3 * sin(53°)) * t - (1/2) * 9.8 * t^2
Simplifying the equation, we get:
4.9t^2 + (21.3 * sin(53°))t - 3.05 = 0
Using this quadratic equation, we can solve for t.
Once we have the value of t, we can use it to find the horizontal displacement (distance from the goal) using the horizontal component of the motion. We'll use the equation:
x = x0 + v0x * t
Where:
x = horizontal displacement (distance from the point of kick to the goal) = 31.2 m
x0 = initial horizontal position = 0 m
v0x = initial horizontal velocity = v0 * cos(θ)
t = time
Substituting the known values:
31.2 = 0 + (21.3 * cos(53°)) * t
Simplifying the equation, we get:
31.2 = (21.3 * cos(53°))t
Solving for t, we can substitute it back into the equation for vertical displacement (y) to find how much the ball clears or falls short of the crossbar.