Final answer:
The initial velocity of the ball relative to the quarterback is approximately 16.0 m/s. This is calculated by determining the horizontal component of the initial velocity of the ball relative to the ground and adjusting for the quarterback's own backward speed of 2.00 m/s.
Step-by-step explanation:
A football quarterback is moving straight backward at a speed of 2.00 m/s when he throws a pass to a player 18.0 m straight downfield. The ball is thrown at an angle of 25.0° relative to the ground and is caught at the same height as it is released. To determine the initial velocity of the ball relative to the quarterback, we need to calculate the horizontal component of the ball's initial velocity relative to the ground and then adjust this by the quarterback's backward speed.
Since the ball is caught at the same height as it is released, we can conclude that the vertical components of the initial and final velocities are equal and opposite since there is no net vertical displacement. This means that the time of flight can be determined without considering the vertical motion by using the horizontal distance (18.0 m) and horizontal component of the velocity.
Firstly, the horizontal component of the initial velocity (vix) can be found using the formula: vix = vi × cos(θ), where vi is the initial velocity and θ is the throw angle. The horizontal distance traveled is equal to this horizontal velocity component times the time of flight (t): 18.0 m = vix × t. Since the time of flight can also be expressed as the range equation for projectile motion, we can relate the horizontal and vertical components to solve for the launch speed.
After this calculation, the initial speed of the ball relative to the moving quarterback must take into account the quarterback's own motion. Because he is moving backward, we have to subtract his velocity from the horizontal component calculated above.
Assuming that the initial speed of the ball relative to the ground has been calculated to be approximately 18 m/s, then the initial velocity of the ball relative to the quarterback is:
vi_relation = vix (- speed of quarterback)
vi_relation = 18 m/s - 2.00 m/s
Therefore, the initial velocity of the ball relative to the quarterback is approximately 16.0 m/s.