Final answer:
To find the maximum speed the kicker can impart to the football, we can use the principle of projectile motion. By calculating the time of flight, horizontal velocity, and using the Pythagorean theorem to find the resultant velocity, we can determine that the maximum speed is approximately 58.6 m/s.
Step-by-step explanation:
To find the maximum speed the field goal kicker can impart to the football, we can use the principle of projectile motion. When the football clears the 3-m-high crossbar, its vertical displacement is equal to the crossbar's height. So we can use the kinematic equation: Δy = V₀y*t + (1/2)*a*t², where Δy is the vertical displacement, V₀y is the initial vertical velocity, t is the time of flight, and a is the acceleration due to gravity. Since the football's initial vertical velocity is zero and the acceleration due to gravity is constant, the equation simplifies to Δy = (1/2)*a*t². Rearranging the equation to solve for t gives us: t = √(2*Δy/a). Plugging in the values, we get: t = √(2*3/9.8) = √(0.61) ≈ 0.78 s.
Next, we can use the horizontal distance and time of flight to find the horizontal velocity V₀x. The equation to use is: Δx = V₀x*t. Rearranging the equation to solve for V₀x gives us: V₀x = Δx/t. Plugging in the values, we get: V₀x = 45.7 m / 0.78 s ≈ 58.6 m/s.
Finally, we can find the maximum speed the kicker can impart to the football using the horizontal and vertical components of velocity. The maximum speed is equal to the magnitude of the resultant velocity vector, which can be found using the Pythagorean theorem: V_max = √(V₀x² + V₀y²). Plugging in the values, we get: V_max = √((58.6)² + (0)²) = √(3433) ≈ 58.6 m/s.