A person going for a morning jog on the deck of a cruise ship is running toward the bow (front) of the ship at 2.5 m/s while the ship is moving ahead at 8.8 m/s. What is the velocity of the jogger relative to the water? Later, the jogger is moving toward the stern (rear) of the ship. What is the jogger's velocity relative to the water now? 

Verified step by step guidance

1. Understand the concept of relative velocity: The velocity of an object relative to a reference frame is the vector sum of the object's velocity relative to another reference frame and the velocity of that reference frame relative to the first reference frame. Here, the jogger's velocity relative to the water is the sum of the jogger's velocity relative to the ship and the ship's velocity relative to the water.
2. Define the given quantities: The velocity of the jogger relative to the ship is \( v_{j/s} = 2.5 \; \text{m/s} \) (toward the bow in the first case and toward the stern in the second case). The velocity of the ship relative to the water is \( v_{s/w} = 8.8 \; \text{m/s} \) (forward).
3. For the first case (jogger running toward the bow): Use the relative velocity equation \( v_{j/w} = v_{j/s} + v_{s/w} \). Since both velocities are in the same direction (toward the bow), add the magnitudes of \( v_{j/s} \) and \( v_{s/w} \).
4. For the second case (jogger running toward the stern): Use the same relative velocity equation \( v_{j/w} = v_{j/s} + v_{s/w} \). However, this time, the jogger's velocity relative to the ship \( v_{j/s} \) is in the opposite direction to the ship's velocity relative to the water \( v_{s/w} \). Subtract the magnitude of \( v_{j/s} \) from \( v_{s/w} \).
5. Summarize the results: The jogger's velocity relative to the water in the first case is the sum of the two velocities, and in the second case, it is the difference between the ship's velocity and the jogger's velocity. Ensure the direction of the resulting velocity is clearly stated in each case.

Key Concepts

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular reference frame. In this scenario, the jogger's velocity is measured relative to the water, which serves as the reference frame. To find the jogger's velocity relative to the water, one must consider both the jogger's speed and the ship's speed, adding or subtracting them based on the direction of motion.

Vector Addition

Vector addition is a mathematical operation used to combine two or more vectors to determine a resultant vector. In this case, the velocities of the jogger and the ship are vectors that can be added or subtracted depending on their directions. When the jogger runs in the same direction as the ship, their velocities add; when running in the opposite direction, they subtract.

Frame of Reference

A frame of reference is a coordinate system used to measure the position and motion of objects. In this problem, the water serves as the frame of reference for calculating the jogger's velocity. Understanding the frame of reference is crucial, as it determines how velocities are perceived and calculated, influencing the final results of the jogger's speed relative to the water.