In Canadian football, after a touchdown the team has the opportunity to earn one more point by kicking the ball over the bar between the goal posts. Football regulations are stated in British units (for which g = 32.26 ft/s2): the bar is 10 feet (3.05 m) above the ground, and the ball is kicked from ground level, 32 yards (29.26 m) horizontally from the bar (Fig. P3.60). (a) There is a minimum angle above the ground such that if the ball is launched below this angle, it can never clear the bar, no matter how fast it is kicked. What is this angle? (b) If the ball is kicked at 45.0° above the horizontal, what must its initial speed be if it is just to clear the bar? Express your answer in m/s and in km/h
To determine the minimum angle above the ground, we need to find the maximum height the ball can reach. At this height, the ball must still be able to clear the 10-foot bar.
(a) To find the minimum launch angle, we can use the following steps:
Step 1: Calculate the maximum height reached by the ball.
We can use the kinematic equation for vertical motion:
h = (v₀ * sin(θ))² / (2 * g)
Where:
h = maximum height reached by the ball
v₀ = initial velocity of the ball
θ = launch angle
g = acceleration due to gravity (32.26 ft/s²)
Step 2: Calculate the time it takes for the ball to reach the maximum height.
We can use the kinematic equation for vertical motion:
v = v₀ * sin(θ) - g * t
Where:
v = final velocity (0 m/s at maximum height)
v₀ = initial velocity of the ball
θ = launch angle
g = acceleration due to gravity (32.26 ft/s²)
t = time taken to reach the maximum height
Since v = 0, we can rewrite the equation as:
0 = v₀ * sin(θ) - g * t
Solving for t, we get:
t = v₀ * sin(θ) / g
Step 3: Calculate the horizontal distance traveled by the ball.
We can use the kinematic equation for horizontal motion:
d = v₀ * cos(θ) * t
Where:
d = horizontal distance traveled by the ball
v₀ = initial velocity of the ball
θ = launch angle
t = time taken to reach the maximum height
Step 4: Set up the constraints for clearing the bar.
The minimum condition for clearing the bar is:
h > 10 feet
Converting 10 feet to meters, we get:
h > 3.05 m
Substituting the equations from Steps 1, 2, and 3 into the constraint equation, we can solve for the minimum angle above the ground.
(b) To find the initial speed required at a launch angle of 45.0° to clear the bar, we need to use the same steps as above, but with the constraint equation:
h = 3.05 m
We can solve for the initial speed by substituting the known values for d, h, and θ into the equations from Steps 1, 2, and 3.
Once we have the initial speed in m/s, we can convert it to km/h by multiplying by 3.6.
By following these steps, we can find the answers to both parts (a) and (b) with the given information.