Problem 57 - Physics for Scientists and Engineers with Modern Physics

Question

(1) A person going for a morning jog on the deck of a cruise ship is running toward the bow (front) of the ship at 2.0 \(\mathrm{m} / \mathrm{s}\) while the ship is moving ahead at 8.5 \(\mathrm{m} / \mathrm{s}\) . What is the velocity of the jogger relative to the water? Later, the joger is moving toward the stern (rear) of the ship. What is the jogger's velocity relative to the water now?

Step-by-Step Solution

Answer

The velocity of the jogger relative to the water is 10.5 m/s toward the bow and 6.5 m/s toward the stern.

Step 1: Understand the Problem

You have a jogger running on a cruise ship moving in the same direction. We want to find the jogger's velocity relative to the water in two scenarios: when jogging towards the bow and towards the stern.

Step 2: Define the Variables and Units

Let the velocity of the ship relative to the water be \( v_s = 8.5 \, \text{m/s} \). The velocity of the jogger relative to the ship is \( v_j = 2.0 \, \text{m/s} \). The velocities are given in meters per second (m/s).

Step 3: Calculate the Velocity Toward the Bow

When the jogger is running towards the bow, both the jogger and the ship are moving in the same direction. The jogger's velocity relative to the water \( v_{bw} \) is the sum of the velocity of the jogger relative to the ship and the velocity of the ship. Thus, \( v_{bw} = v_j + v_s = 2.0 \, \text{m/s} + 8.5 \, \text{m/s} = 10.5 \, \text{m/s} \).

Step 4: Calculate the Velocity Toward the Stern

When the jogger is running towards the stern, they are moving in the opposite direction of the ship. The velocity of the jogger relative to the water \( v_{sw} \) is the velocity of the ship minus the velocity of the jogger. So, \( v_{sw} = v_s - v_j = 8.5 \, \text{m/s} - 2.0 \, \text{m/s} = 6.5 \, \text{m/s} \).

Key Concepts

Kinematics

Kinematics deals with the description of motion without considering the forces that cause motion. It's a fundamental aspect of physics that studies the motion of objects through different mediums, such as the jogger on a moving ship deck in this situation. By focusing on quantities like velocity, displacement, acceleration, and time, kinematics provides valuable insights into the patterns of motion.
In this exercise, two key measurements are presented: the velocity of the ship and the velocity of the jogger. Since both entities are in motion, understanding their velocities is essential for solving the problem. Kinematic principles guide us to explore how these velocities combine or counter each other depending on the jogger's direction of motion.
Thus, kinematics provides the framework by which the problem is approached, ensuring that these relationships and calculations are accurate.

Velocity Addition

Velocity addition is the process of combining two velocities in context, a crucial tool when dealing with multiple moving objects. It is particularly helpful when determining the net velocity of an object seen from another moving object. In this case, understanding velocity addition allows us to calculate the jogger's overall velocity relative to the stationary water.
When the jogger moves towards the bow, both the jogger's and ship's velocities are added together. This is because both are moving in the same direction, thus their velocities reinforce each other.

• The calculation was: \( v_{bw} = v_j + v_s = 10.5 \text{ m/s} \)

Similarly, when the jogger moves towards the stern, velocity addition considers that the velocities are in opposite directions. The calculation changes slightly:

• Resulting in: \( v_{sw} = v_s - v_j = 6.5 \text{ m/s} \)

Overall, velocity addition is key to understanding and accurately determining relative motion in various scenarios.

Reference Frames

Reference frames are a fundamental concept in physics that simplify the understanding of motion. A reference frame is essentially a point of view from which measurements and observations are made. When an object is observed from different reference frames, it may have different apparent velocities or movements.
In this exercise, two reference frames are particularly relevant:

• The ship's frame, which sees the jogger moving at 2.0 m/s.


• The water's frame, which observes the combined effect of the ship's velocity and the jogger's velocity.

The perspective of the stationary water is crucial when determining the jogger's velocity relative to it, as the ship itself is already in motion relative to the water. By choosing the reference frame carefully, one can simplify the complexity of the problem, effectively making relative velocity problems, like this one, easier to solve.
Understanding reference frames enriches our comprehension of how motion is perceived from different viewpoints, offering clarity in both theoretical and practical applications.