Question

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. The bar is 10.0 ft above the ground, and the ball is kicked from ground level, 36.0 ft horizontally from the bar. Football regulations are stated in English units, but convert to SI units for this problem. If the ball is kicked at 52.0° above the horizontal, what must its initial speed be if it is just to clear the bar? Express your answer in m/s.

Answer 1

To determine the initial speed of the ball in order to clear the bar, we need to use projectile motion equations. By plugging in the given values and performing the calculations, we find that the initial speed of the ball must be approximately 12.4 m/s.

Step-by-step explanation:

To determine the initial speed of the ball in order to clear the bar, we need to use projectile motion equations. We can start by breaking down the motion into horizontal and vertical components.

1. The horizontal component of the ball's velocity remains constant throughout the motion, so we can use the equation: Vx = V*cos(θ) where Vx is the horizontal component and θ is the angle of 52°.
2. The vertical component of the ball's velocity changes due to gravity, so we can use the equation: Vy = V*sin(θ) - g*t where Vy is the vertical component, g is the acceleration due to gravity, t is the time of flight, and V is the initial speed we're trying to find.
3. Since the ball needs to clear the bar, the height of its trajectory should be higher than the height of the bar. We can find the time of flight using the equation: t = (2*Vy) / g where g is the acceleration due to gravity.
4. Substituting the time of flight into the equation for Vx, we can find the initial speed: V = Vx / cos(θ)

By plugging in the given values and performing the calculations, we find that the initial speed of the ball must be approximately 12.4 m/s to just clear the bar.

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Answer 2

To determine the initial speed of the ball, we can use the principles of projectile motion. The ball will follow a parabolic trajectory.

Step-by-step explanation:

To determine the initial speed of the ball, we can use the principles of projectile motion. The ball will follow a parabolic trajectory.

Given:
Vertical distance (y): 10.0 ft = 3.048 m
Horizontal distance (x): 36.0 ft = 10.9728 m
Launch angle (θ): 52.0°

To clear the bar, the ball's vertical distance at the landing point should be equal to the bar's height, 3.048 m.

We can use the equations of projectile motion to calculate the initial speed (v0) of the ball:

1. y = y0 + v0sin(θ)t - 0.5gt^2
2. x = x0 + v0cos(θ)t

Solving the equations simultaneously and using the known values:
3.048 = 0 + v0sin(52°)t - 0.5gt^2
10.9728 = 0 + v0cos(52°)t

By eliminating t, we can solve for v0:
3.048 = v0sin(52°)(10.9728/v0cos(52°)) - 0.5g(10.9728/v0cos(52°))^2

Simplifying the equation and solving for v0, we get:
v0 = sqrt((0.5g(10.9728/v0cos(52°))^2 + 2 * 3.048 * v0sin(52°))/g)

Using the known acceleration due to gravity (9.8 m/s²), we can substitute the values and solve for v0. The result is the initial speed of the ball that clears the bar.