A football kicker can give the ball an initial speed of 25 m/s. What are the (a) least and (b) greatest elevation
angles at which he can kick the ball to score a field goal from a point 50 m in front of goalposts whose
horizontal bar is 3.44 m above the ground?
To determine the least and greatest elevation angles at which the football kicker can kick the ball to score a field goal, we need to analyze the projectile motion of the ball.
The given information includes:
- Initial speed of the ball (v₀) = 25 m/s
- Distance from the point of the kicker to the goalposts (horizontal distance) (d) = 50 m
- Height of the goalposts' horizontal bar (h) = 3.44 m
Let's start by breaking down the problem into components.
Horizontal Motion:
The horizontal motion of the ball is independent of its vertical motion. The horizontal distance traveled by the ball (d) is given as 50 m.
Vertical Motion:
The vertical motion of the ball can be analyzed using the equations of projectile motion.
We can use the following kinematic equations in vertical motion, considering the initial vertical velocity (v₀sinθ) and the acceleration due to gravity (g = 9.8 m/s²):
1. Vertical displacement (Δy) = v₀sinθ * t - (1/2) * g * t²
2. Final vertical velocity (v) = v₀sinθ - g * t
3. Time of flight (T) = (2 * v₀sinθ) / g
Now, let's solve for the least and greatest elevation angles (θ).
(a) Least Elevation Angle:
To find the least elevation angle, we need to determine the angle at which the ball just clears the goalposts' horizontal bar (h = 3.44 m).
Using Equation (1), we can substitute Δy as h = 3.44 m and solve for t. Rearranging the equation, we get:
h = v₀sinθ * t - (1/2) * g * t²
Rearranging further:
(1/2) * g * t² + v₀sinθ * t - h = 0
This equation is in the form of a quadratic equation in terms of t. Solving for t using the quadratic formula gives us two possible values of t. We can ignore the negative solution since time cannot be negative in this context.
After obtaining the value of t, we can substitute it into Equation (3) to find the corresponding value of θ.
(b) Greatest Elevation Angle:
To find the greatest elevation angle, we need to determine the angle at which the ball lands exactly at the horizontal distance of 50 m from the kicker.
Similar to the least elevation angle, we need to determine the time of flight (T) using Equation (3). Then, we can substitute T into Equation (1) to solve for the maximum height reached by the ball (Δy).
Using the calculated Δy, we can calculate the corresponding angle θ using Equation (1).
By solving these equations, we can find the least and greatest elevation angles at which the kicker can score a field goal, given the provided information.