A quarterback claims that he can throw the football a horizontal distance of 187 m. Furthermore, he claims that he can do this by launching the ball at the relatively low angle of 32.4 ° above the horizontal. To evaluate this claim, determine the speed with which this quarterback must throw the ball. Assume that the ball is launched and caught at the same vertical level and that air resistance can be ignored. For comparison a baseball pitcher who can accurately throw a fastball at 45 m/s (100 mph) would be considered exceptional.

hf=hi+v*sinTheta-g/2 t^2

but
187=vcosTheta*t solve for t here, put it in the first equation, solve for v.

have fun.

To determine the speed with which the quarterback must throw the ball, we can use the principles of projectile motion. The horizontal distance covered by the football depends on two factors: the initial speed of the throw and the launch angle.

We'll start by breaking down the initial velocity into horizontal and vertical components. The horizontal component remains constant throughout the projectile's motion, while the vertical component changes due to gravity.

Given:
- Range (horizontal distance) = 187 m
- Launch angle (θ) = 32.4°

Step 1: Calculate the horizontal component of the initial velocity.
The horizontal component (Vx) of the initial velocity can be found using trigonometry:
Vx = V × cos(θ)

Step 2: Calculate the vertical component of the initial velocity.
To find the vertical component (Vy) of the initial velocity, we can use the equation:
Vy = V × sin(θ)

Step 3: Determine the time of flight.
The time of flight (t) is the total time it takes for the projectile to reach the same vertical level where it was launched.
t = 2 × (Vy / g), where g is the acceleration due to gravity (9.8 m/s²).

Step 4: Calculate the initial speed.
Using the formula for the range of a projectile motion:
Range = Vx × t
Solving for Vx:
Vx = Range / t

Step 5: Substitute the calculated values into the equations to find the initial speed (V).
V = Vx / cos(θ)

Now we can plug in the known values and calculate the initial speed (V):

Step 1: Calculate Vx:
Vx = V × cos(32.4°)

Step 2: Calculate Vy:
Vy = V × sin(32.4°)

Step 3: Determine the time of flight:
t = 2 × (Vy / g)

Step 4: Calculate Range:
Range = Vx × t

Step 5: Solve for V:
V = Range / t / cos(θ)

By performing these calculations, we can find the required speed with which the quarterback must throw the ball to achieve a range of 187 m at a launch angle of 32.4° above the horizontal. Note that air resistance is assumed to be negligible in this scenario.