A quarterback throws the football to a stationary receiver who is 19.7 m down the field. The football is thrown at an initial angle of 43◦ to the ground.

The acceleration of gravity is 9.81 m/s^2.
a) At what initial speed must the quarterback throw the ball for it to reach the receiver?
Answer in units of m/s.
b) What is the ball’s highest point during its flight?
Answer in units of m.

a) To find the initial speed at which the quarterback must throw the ball for it to reach the receiver, we can analyze the motion of the football using projectile motion equations.

We can break down the given information as follows:
- Initial vertical displacement (y) = 19.7 m
- Initial angle (θ) = 43°
- Acceleration due to gravity (g) = 9.81 m/s^2

Using the vertical motion equation for the maximum height (H) reached by the ball during its flight:

Δy = V₀ * sin(θ) * t - (1/2) * g * t²

Where:
- Δy is the change in vertical displacement (maximum height)
- V₀ is the initial vertical velocity
- θ is the initial angle of the throw
- t is the time taken to reach the maximum height

At the maximum height, the vertical velocity becomes zero, so V_y = V₀ * sin(θ) - g * t = 0. Solving for t, we get:

t = V₀ * sin(θ) / g

Now, let's find the time taken to reach the receiver (T):

T = (2 * t)

The horizontal distance (x) traveled by the ball can be calculated using the equation:

x = V₀ * cos(θ) * T

The horizontal displacement (x) is equal to the distance down the field, so:

x = 19.7 m

We have two unknowns in the equations above: V₀ (initial speed) and T (total time). We can solve these equations simultaneously using the given values to find the initial speed (V₀).

Plugging in the known values:

19.7 m = V₀ * cos(43°) * (2 * (V₀ * sin(43°) / g))

Simplifying the equation:

19.7 = V₀² * sin(43°) * cos(43°) / g

To isolate V₀, we rearrange the equation:

V₀² = 19.7 * g / (sin(43°) * cos(43°))

V₀ = sqrt(19.7 * g / (sin(43°) * cos(43°)))

Evaluating the expression on the right side, we can find the value of V₀.

b) To find the maximum height (H) reached by the ball during its flight, we can use the equation for the vertical displacement (y) at any given time (t):

y = V₀ * sin(θ) * t - (1/2) * g * t²

At the maximum height, the vertical velocity is zero: V_y = V₀ * sin(θ) - g * t = 0. Solving for t, we get:

t = V₀ * sin(θ) / g

Substituting this value of t back into the equation for y, we can find the maximum height:

H = V₀ * sin(θ) * (V₀ * sin(θ) / g) - (1/2) * g * (V₀ * sin(θ) / g)²

Simplifying the equation, we can calculate the value of H.

H = (V₀² * sin²(θ)) / (2 * g)