Problem 57 - Physics for Scientists and Engineers with Modern Physics

Question

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 jogger 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 jogger's velocity relative to the water is \(10.5 \mathrm{~m/s}\) toward the bow and \(6.5 \mathrm{~m/s}\) toward the stern.

Step 1: Define Relative Velocity

When considering the velocity of the jogger relative to the water, we must add the velocity of the jogger relative to the ship to the velocity of the ship relative to the water. The velocities are vectors, so their directions must be taken into account.

Step 2: Calculate Velocity Towards the Bow

The jogger is running toward the bow of the ship at a speed of \(2.0 \mathrm{~m/s}\). The velocity of the ship relative to the water is \(8.5 \mathrm{~m/s}\) in the same direction. The total velocity of the jogger relative to the water is the sum: \(2.0 \mathrm{~m/s} + 8.5 \mathrm{~m/s} = 10.5 \mathrm{~m/s}\).

Step 3: Calculate Velocity Towards the Stern

When the jogger is moving toward the stern of the ship, their velocity relative to the ship becomes \(-2.0 \mathrm{~m/s}\). The negative sign indicates they are moving in the opposite direction of the ship's movement. Thus, the jogger's velocity relative to the water is \(8.5 \mathrm{~m/s} - 2.0 \mathrm{~m/s} = 6.5 \mathrm{~m/s}\).

Key Concepts

Velocity Calculation

Calculating velocity involves determining how fast an object moves and in which direction. It's a vector quantity, meaning both speed and direction are crucial. To calculate relative velocity, you often have to combine different velocities based on their directions. For example, if you know your own speed and the speed of the moving surface you're on, you can add or subtract these to find your speed relative to an outside observer. In our example, the jogger's speed on a ship was combined with the ship's speed to determine his velocity relative to the water. By adding when moving in the same direction and subtracting when moving in opposite directions, one can find the jogger's net velocity relative to the water.

Reference Frames

Understanding reference frames is crucial for solving problems involving relative velocity. A reference frame is a perspective from which you measure and observe motion. In the jogger's scenario, different reference frames help explain the relative motion:

• **Jogger's Reference Frame:** From the perspective of the jogger, their speed is either towards the bow or away from it, at a set pace.
• **Ship's Reference Frame:** The ship moves at 8.5 m/s relative to the water and affects the jogger's movement from an outside viewer's perspective.
• **Water's Reference Frame:** Here, we measure the actual speed of the jogger relative to a stationary point, the water.

By understanding the different frames, we observe how the jogger's velocity interacts and changes when viewed from another point.

Vector Addition

Vector addition is a key process when dealing with multiple velocities that have both magnitude and direction. Vectors are typically represented as arrows, where the length represents magnitude and the direction of the arrow indicates direction. You perform vector addition by placing the tail of one vector to the tip of another. In the case of the jogger:

• If the jogger moves towards the bow, the vectors of both the jogger and the ship point in the same direction. Their magnitudes are added directly: 2.0 m/s (jogger) + 8.5 m/s (ship) = 10.5 m/s.
• If moving towards the stern, the jogger's vector is in the opposite direction of the ship's vector, hence the jogger's vector is negative: 8.5 m/s (ship) - 2.0 m/s (jogger) = 6.5 m/s.

This technique helps to easily determine resulting velocities in different scenarios.