A quarterback throws a football toward a receiver with an initial speed of 22 m/s at an angle of 30° above the horizontal. At that instant the receiver is 23.6 m from the quarterback.

With what constant speed should the receiver run in order to catch the football at the level at which it was thrown?

To find the constant speed at which the receiver should run to catch the football at the level it was thrown, we can use the kinematic equations of motion and the principles of projectile motion.

Step 1: Resolve the initial velocity into horizontal and vertical components.
The initial velocity (V0) of the football can be resolved into horizontal (V0x) and vertical (V0y) components using the given angle of 30° and the trigonometric relationships. The horizontal component remains constant throughout the flight, whereas the vertical component is affected by gravity.

V0x = V0 * cos(θ)
V0y = V0 * sin(θ)

Given:
V0 = 22 m/s
θ = 30°

V0x = 22 * cos(30°)
V0x = 22 * 0.866
V0x ≈ 19.0 m/s

V0y = 22 * sin(30°)
V0y = 22 * 0.5
V0y ≈ 11.0 m/s

Step 2: Calculate the time of flight.
The time of flight (t) is the total time it takes for the football to reach the receiver. Since the initial and final vertical positions are the same, we can use the vertical component of the motion to calculate the time.

Using the equation:
Δy = V0y * t + (1/2) * a * t^2

where:
Δy = 0 (since final vertical position is same as initial)
V0y = initial vertical velocity component
a = acceleration due to gravity (approximately -9.8 m/s^2)
t = time of flight

0 = 11.0 * t + (1/2) * (-9.8) * t^2

Simplifying the equation:
-4.9t^2 + 11t = 0

Factoring out t:
t(-4.9t + 11) = 0

t = 0 (initial time is not considered)
or
-4.9t + 11 = 0

Solving for t:
-4.9t = -11
t = -11 / -4.9
t ≈ 2.24 s

Step 3: Calculate the horizontal distance covered by the receiver.
The horizontal distance covered by the receiver is the product of the constant speed (Vr) and the time of flight (t).

Given:
Vr = Unknown constant speed
t = 2.24 s

Horizontal distance (d) = speed * time
23.6 m = Vr * 2.24

Solving for Vr:
Vr = 23.6 / 2.24
Vr ≈ 10.54 m/s

Therefore, the receiver should run at a constant speed of approximately 10.54 m/s to catch the football at the level it was thrown.