Problem 59 - College Physics

Question

A container ship is traveling westward at a speed of 5.00 \(\mathrm{m} / \mathrm{s} .\) The waves on the surface of the ocean have a wave-length of 40.0 \(\mathrm{m}\) and are traveling eastward at a speed of 16.5 \(\mathrm{m} / \mathrm{s} .\) (a) At what time intervals does the ship encounter the crest of a wave? (b) At what time intervals will the ship encounter wave crests if it turns around and heads eastward?

Step-by-Step Solution

Answer

(a) Approx. 1.86 s; (b) Approx. 3.48 s.

Step 1: Understanding Relative Velocity

First, understand the concept of relative velocity. The ship and the waves are on a collision course due to their opposing directions. Therefore, we must compute the relative speed to determine how often the ship encounters wave crests. The relative speed of the ship with respect to the waves is the sum of their individual speeds because they are moving in opposite directions.

Step 2: Calculate Relative Velocity for Part (a)

For part (a), the ship is moving westward at 5.00 m/s and the waves are moving eastward at 16.5 m/s. The relative velocity is the sum of these speeds:\[V_{\text{relative}} = V_{\text{waves}} + V_{\text{ship}} = 16.5 \, \text{m/s} + 5.00 \, \text{m/s} = 21.5 \, \text{m/s}.\]

Step 3: Compute Time Interval for Wave Encounters (Part a)

The time interval for the ship encountering a wave crest is determined by dividing the wavelength by the relative velocity:\[T = \frac{\lambda}{V_{\text{relative}}} = \frac{40.0 \, \text{m}}{21.5 \, \text{m/s}} \approx 1.86 \, \text{s}.\]Hence, the ship encounters wave crests every approximately 1.86 seconds.

Step 4: Calculate Relative Velocity for Part (b)

For part (b), now assume the ship is heading eastward, the same direction as the waves. The relative velocity is the difference in speeds since both are moving in the same direction:\[V_{\text{relative}} = V_{\text{waves}} - V_{\text{ship}} = 16.5 \, \text{m/s} - 5.00 \, \text{m/s} = 11.5 \, \text{m/s}.\]

Step 5: Compute Time Interval for Wave Encounters (Part b)

Now compute the time interval for encountering wave crests:\[T = \frac{\lambda}{V_{\text{relative}}} = \frac{40.0 \, \text{m}}{11.5 \, \text{m/s}} \approx 3.48 \, \text{s}.\]Therefore, the ship encounters wave crests every approximately 3.48 seconds when traveling eastward.

Key Concepts

Understanding Wavelength

A wavelength is a fundamental concept in physics, especially when studying waves, such as those on the ocean. It represents the distance between two successive crests (the highest points) or troughs (the lowest points) of a wave. When considering the problem of a ship encountering wave crests, the wavelength gives us crucial information about how these crests are spaced out along the ocean's surface.
For our exercise, the wavelength is 40 meters. This means if you were to stand still and watch the waves move past, you would see a crest every 40 meters. In ocean navigation, understanding the wavelength helps predict when and how often waves will impact a moving object, such as a ship.

Analyzing Time Intervals

Time intervals refer to the amount of time it takes for a repeating event to occur. In our exercise, time intervals are used to determine how frequently the ship hits a wave crest.
When the ship moves westward, its speed relative to the moving waves needs to be calculated to find out these time intervals. By using the formula \( T = \frac{\lambda}{V_{\text{relative}}} \), where \( \lambda \) is the wavelength and \( V_{\text{relative}} \) is the relative velocity, we can determine the frequency of ship-wave encounters.
For example, with a relative velocity of 21.5 m/s (when the ship moves west), the time interval is approximately 1.86 seconds. This means the ship hits a new wave crest every 1.86 seconds.

On a Collision Course

When two objects are on a collision course, they are moving directly toward each other from opposing directions. In this scenario, the concept is observed with the ship and the ocean waves.
Since the ship is moving westward at 5.00 m/s and the waves are moving eastward at 16.5 m/s, the relative velocity is computed by adding their speeds: \( V_{\text{relative}} = 16.5 + 5.00 = 21.5 \) m/s.
This increased speed indicates how fast the ship and waves approach each other, allowing us to compute when the ship will hit a crest. When the ship changes direction and goes east, it follows the waves, reducing the relative velocity to 11.5 m/s, increasing time intervals for wave encounters.

The Dynamics of Wave Motion

Wave motion describes how energy travels through a medium, such as water, without significant transport of matter. In the context of ocean waves, this motion is critical to understanding how waves impact ships.
Ocean waves move toward the shore, causing periodic motion. Knowing the speed of these waves, specified as 16.5 m/s in our problem, helps calculate how frequently they meet a ship. The ship's direction and speed also affect this interaction, modifying the perceived frequency due to the Doppler effect.
When waves pass by a vessel, energy is transferred, causing the sensation of rising and falling. Understanding wave motion can guide ship captains in plotting safer routes and ensuring smoother travels across stormy seas.