Final answer:
The angle required for the ball to be caught by the running woman is found using projectile motion concepts, including the horizontal and vertical components of velocity. The time 't' during which the ball is in the air and the distance the woman covers determine the necessary horizontal velocity component. Calculating these values leads us to the angle at which the ball must be thrown.
Step-by-step explanation:
To solve for the angle at which the ball should be thrown for the runner to catch it, we need to use the principles of projectile motion. The problem can be analyzed in two components: horizontal motion and vertical motion. Since there's no air resistance, the horizontal component of the ball's velocity will remain constant throughout its flight.
The time it takes for the ball to hit the ground is determined by the vertical motion. Using the formula t = √(2h/g), where ‘h’ is the height of the cliff and ‘g’ is the acceleration due to gravity, we can calculate the time the ball is in the air. With a height of 45.0 m and ‘g’ being 9.8 m/s², the time 't' can be calculated.
Since the woman is running away at a speed of 6.00 m/s, the distance she covers is the product of her speed and the time 't'. We can then find the horizontal component of the ball's velocity needed to cover the same distance.
The initial speed of the ball has both a horizontal component (vx) and a vertical component (vy). The angle θ can be found using trigonometric relations: vx = v*cos(θ) and vy = v*sin(θ) where 'v' is the initial speed. By equating the horizontal component with the product of the runner's speed and the time 't', and knowing the initial speed of the ball is 20.0 m/s, we can solve for the angle θ.