To find the initial speed at which the quarterback must throw the ball, we can use the kinematic equation for the horizontal distance traveled:
d = v₀ * t
where d is the horizontal distance (18.3 m) and t is the time it takes for the ball to reach the receiver.
To find the time, we need to break down the initial velocity of the ball into its horizontal and vertical components. We can use the equation:
v₀x = v₀ * cosθ
v₀y = v₀ * sinθ
where v₀x is the horizontal component of the initial velocity, v₀y is the vertical component of the initial velocity, and θ is the angle at which the ball was thrown (44.9 degrees).
The horizontal distance traveled can be found using the equation:
d = v₀x * t
where t is the time it takes for the ball to reach the receiver.
From the equations above, we can substitute the values to get:
18.3 m = (v₀ * cos44.9) * t
Since we have two unknowns (v₀ and t) and only one equation, we need another equation to solve for both variables.
Next, let's consider the vertical motion of the ball.
The equation for the vertical distance traveled is given by:
y = v₀y * t + (1/2) * a * t²
where y is the vertical distance traveled (ending position - starting position), v₀y is the vertical component of the initial velocity, a is the acceleration due to gravity (-9.81 m/s²), and t is the time.
Since the ball reaches its highest point at the midpoint of its flight, the vertical distance traveled at this point is 0. Therefore, we can set y = 0. Substituting the values:
0 = (v₀ * sin44.9) * t + (1/2) * (-9.81) * t²
Now, we have two equations:
18.3 = (v₀ * cos44.9) * t (Equation 1)
0 = (v₀ * sin44.9) * t + (1/2) * (-9.81) * t² (Equation 2)
We can solve these equations simultaneously to find the values of v₀ and t.
Once we have the value of t, we can substitute it back into Equation 1 to find the initial speed (v₀).
To find the highest point during the ball's flight, we need to calculate the time it takes for the ball to reach that point. At the highest point, the vertical velocity of the ball is momentarily zero. We can find this time using the equation:
v = v₀y + a * t
Setting v = 0, we can solve for t:
0 = (v₀ * sin44.9) + (-9.81) * t
Solving for t will give us the time it takes for the ball to reach its highest point. We can then substitute this value into the equation for the vertical distance traveled to find the height at the highest point.