A quarterback throws a football toward a receiver with an initial speed of 21 m/s at an angle of 30° above the horizontal. At that instant, the receiver is 12 m from the quarterback. In what direction and with what constant speed should the receiver run in order to catch the football at the level at which it was thrown?
I keep getting 14.58, but it's incorrect.
The distance the ball travels to reach the level it was thrown is
[V^2/(g)] sin (2A) = 39 m
The receiver must run a distance of 39-12 = 27 m during the time the ball is in the air above arm level, which is 2 V sin A/g = 2.14 s. The required speed is 39 m/2.14 s = 18.2 m/s. No human can run that fast.
To solve this problem, we can break it down into horizontal and vertical components. Let's first find the time it takes for the football to reach the receiver.
The initial velocity of the football can be broken down into its horizontal and vertical components:
Vx = V_initial * cos(theta)
Vy = V_initial * sin(theta)
Given that the initial speed, V_initial, is 21 m/s and the angle, theta, is 30°, we can calculate the horizontal and vertical components:
Vx = 21 m/s * cos(30°)
Vy = 21 m/s * sin(30°)
Vx = 21 m/s * √(3)/2
Vx = 10.5√(3) m/s
Vy = 21 m/s * 1/2
Vy = 10.5 m/s
The time, t, it takes for the football to reach the receiver can be found using the vertical component:
Vy = g * t
10.5 m/s = 9.8 m/s^2 * t
t = 10.5 m/s / 9.8 m/s^2
t ≈ 1.07 s
Now, let's determine how far the receiver needs to move horizontally in that time.
The horizontal distance, Dx, can be calculated using the horizontal velocity:
Dx = Vx * t
Dx = 10.5√(3) m/s * 1.07 s
Dx ≈ 11.36√(3) m
Given that the receiver is initially 12 m away from the quarterback, the receiver needs to move Dx - 12 m horizontally to catch the football.
Now, we can calculate the direction and magnitude of the velocity the receiver needs to run.
The angle, alpha, between the required velocity and the horizontal direction can be calculated using the horizontal and vertical displacements:
tan(alpha) = (Dx - 12 m) / 10.5 m
alpha = arctan((Dx - 12 m) / 10.5 m)
The magnitude of the velocity, V_receiver, can be calculated by dividing the horizontal displacement by the time taken:
V_receiver = (Dx - 12 m) / t
Finally, the direction and magnitude the receiver needs to run are:
Direction: alpha (angle above the horizontal)
Magnitude: V_receiver (calculated above)
By plugging in the values calculated earlier, you should be able to determine the correct direction and magnitude for the receiver to catch the football.