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Trajectory analysis refers to the path that a projectile follows through space under the influence of gravitational forces. In this exercise, the ball is thrown from the top of a cliff which means its initial position is elevated, influencing its trajectory. Typically, a projectile follows a parabolic path. To analyze this path, we need to consider both the horizontal and vertical components of the initial velocity, as provided in the statement, and how they contribute to the overall motion of the projectile. The initial velocity of 20.0 m/s can be broken down into components based on the angle of projection. Understanding the projectile's trajectory is crucial because it helps us determine where and when the ball will land, allowing us to synchronize this with the runner's path.

Equations of motion describe how an object's velocity and position change over time due to external forces, such as gravity. For a projectile, we primarily use the following kinematic equations to determine its motion:

• Horizontal motion: - For any point in time, the horizontal distance can be described by the equation: \( x = v_{0x} t \).
• Vertical motion: - The vertical position is altered by gravity, given by the kinematic equation: \( y = h + v_{0y} t - \frac{1}{2}gt^2 \), where \( h \) is the initial height and \( g \) is the acceleration due to gravity.

These equations allow us to calculate how long the projectile will be in motion (time of flight) and how far it will travel horizontally when analyzed together.

When breaking down projectile motion, it's essential to separate the initial velocity into horizontal and vertical components. The horizontal component \( v_{0x} = v_0 \cos(\theta) \) influences how far the projectile will travel, while the vertical component \( v_{0y} = v_0 \sin(\theta) \) affects the projectile's time in the air and the height it can reach. Both components are analyzed independently:

• Horizontal motion has no acceleration (ignoring air resistance), thus the velocity remains constant, impacting how far from the cliff the ball will move horizontally.
• Vertical motion is affected by gravity, which causes the ball to decelerate upward initially and accelerate downward thereafter, impacting when the ball will reach the ground.

Correctly calculating these components of motion is vital for solving for the angle \( \theta \) such that the runner intercepts the projectile at the precise location.

The time of flight is the duration for which a projectile remains in motion from the launch until it lands back on the ground. To determine the time of flight, we solve the vertical motion equation by setting the final vertical position equal to zero (ground level): \( 0 = h + v_{0y} t - \frac{1}{2}gt^2 \). This quadratic equation can yield one or two solutions for \( t \), out of which the positive value of \( t \) is considered as it represents the realistic scenario (i.e., after the launch).In this example:

• Substituting in the values \( h = 45 \), \( v_{0y} = 20 \sin(\theta) \), and \( g = 9.8 \) m/s², you solve for \( t \).
• Incorporating \( \theta \) determined from the horizontal condition helps find the precise time for which the runner's path and the projectile coincide.

Understanding the time of flight helps us synchronize both the ball and the runner to meet just before impact.

Relative motion analysis involves understanding how the motion of the projectile appears from different frames of reference. In this problem, we consider two viewpoints:

• From a stationary observer on the ground, the ball follows a typical parabolic path as expected from projectile motion.
• For the runner, who is moving at the same horizontal speed as the horizontal component of the projectile's motion, the ball appears to move vertically and in a straight path relative to her frame of motion.

To the runner, accounting for her speed of 6.00 m/s, the horizontal relative speed of the ball is zero. Consequently, she perceives the ball as moving directly towards her. Understanding relative motion is crucial in physics as it allows for explaining how different observers may conceptualize the same moving object in various ways.