A place kicker must kick a football from a
point 35.0 m (about 38 yd) from the 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 24.0
m/s at an angle of 51◦
to the horizontal.
To determine if the ball clears the crossbar,
what is its height with respect to the crossbar
when it reaches the plane of the crossbar?
The acceleration of gravity is 9.81 m/s
2
.
Answer in units of m.
find horizontal and vertical launch components
use horizontal component to find flight time (t) to crossbar
use vertical component to find height of football at time (t)
To determine the height of the football with respect to the crossbar when it reaches the plane of the crossbar, we need to split the initial velocity of the ball into its horizontal and vertical components.
Given:
Initial velocity (magnitude): v = 24.0 m/s
Launch angle: θ = 51°
Acceleration due to gravity: g = 9.81 m/s^2
To find the vertical component of the initial velocity, we need to find the sine of the launch angle:
Vertical component = v * sin(θ)
So, the vertical component of the initial velocity is:
Vertical component = 24.0 m/s * sin(51°)
Next, we can find the time it takes for the ball to reach the plane of the crossbar using the vertical component of the velocity and the acceleration due to gravity:
Time = (Vertical component) / (acceleration due to gravity)
Now, we can calculate the height of the ball above the crossbar when it reaches the plane of the crossbar by using the formula for vertical motion:
Height = (Vertical component) * (Time) - (0.5) * (acceleration due to gravity) * (Time^2)
Substituting the known values, we get:
Height = (24.0 m/s * sin(51°)) * [(24.0 m/s * sin(51°)) / (9.81 m/s^2)] - (0.5) * (9.81 m/s^2) * [(24.0 m/s * sin(51°)) / (9.81 m/s^2)]^2
Evaluating this equation will give us the height with respect to the crossbar when the ball reaches the plane of the crossbar.