Problem 68 - College Physics

Question

M Two ships are moving along a line due east (Fig. P14.68). The trailing vessel has a speed relative to a landbased observation point of \(v_{1}=64.0 \mathrm{~km} / \mathrm{h}\), and the leading ship has a speed of \(v_{2}=45.0 \mathrm{~km} / \mathrm{h}\) relative to that point. The two ships are in a region of the ocean where the current is moving uniformly due west at \(v_{\text {current }}=\) \(10.0 \mathrm{~km} / \mathrm{h}\). The trailing ship transmits a sonar signal at a frequency of \(1200.0 \mathrm{~Hz}\) through the water. What frequency is monitored by the leading ship?

Step-by-Step Solution

Answer

The frequency monitored by the leading ship is approximately \(1216.1 \, \text{Hz}\).

Step 1: Determine the relative velocity of the two ships

In this step, compute the relative velocities of the two ships to the water as these must be subtracted to get the relative velocity. The trailing ship is moving faster than the current, so its speed relative to the water is \(v_{1}-v_{\text{current}} = 64.0 \, \text{km/h} - 10.0 \, \text{km/h} = 54.0 \, \text{km/h}\), converted to m/s gives \(15.0 \, \text{m/s}\). The leading ship is moving slower than the current, so its speed relative to the water is \(v_{2}+v_{\text{current}} = 45.0 \, \text{km/h} +10.0 \, \text{km/h} = 55.0 \, \text{km/h}\), converted to m/s gives \(15.3 \, \text{m/s}\).

Step 2: Apply the formula for Doppler effect when both source and observer are moving

The formula for the Doppler effect is \(f'=f(\frac{v+vo}{v+vs})\), where v is the speed of sound in water (1500 m/s), vs is the speed of the source (trailing ship, -15.0 m/s since it's moving in the opposite direction of sound), vo is the speed of the observer (leading ship, 15.3 m/s since it's moving in the same direction of sound), and f is the observed frequency (1200 Hz). Substituting these values into the formula gives \(f'=1200 \, \text{Hz} (\frac{1500 \, \text{m/s} + 15.3 \, \text{m/s}}{1500 \, \text{m/s} - 15.0 \, \text{m/s}})\).

Step 3: Calculate the resulting frequency

By running the calculations in the previous formula, the resulting frequency is found to be around \(1216.1 \, \text{Hz}\). This is the frequency monitored by the leading ship.

Key Concepts

Relative Velocity

When two objects, such as ships, are moving, it's crucial to understand their speeds relative to one another, especially when considering a medium like water. Relative velocity accounts for the movement of the surrounding environment and the objects themselves. In the given scenario:

• The trailing ship moves faster than the current, making its speed relative to the water: \(v_{1} - v_{\text{current}} = 64.0 \, \text{km/h} - 10.0 \, \text{km/h} = 54.0 \, \text{km/h}\).
• This is converted to meters per second: \(15.0 \, \text{m/s}\).
• The leading ship moves slower than the current, so its velocity relative to the water is: \(v_{2} + v_{\text{current}} = 45.0 \, \text{km/h} + 10.0 \, \text{km/h} = 55.0 \, \text{km/h}\).
• This becomes \(15.3 \, \text{m/s}\).

Instead of just looking at the speeds of the ships relative to a fixed point on land, we calculate their speed against the moving current. This changes if both objects are moving towards or away from each other.

Frequency Calculation

To understand how the frequency changes as perceived by the leading ship, we use the Doppler effect formula. This formula helps us calculate the frequency change when the source or observer, or both, are moving. The Doppler effect formula is:\[ f' = f \left(\frac{v + v_o}{v + v_s}\right) \]Where:

• \( f' \) is the observed frequency.
• \( f \) is the emitted frequency (given as 1200 Hz).
• \( v \) is the speed of sound in water (about 1500 m/s).
• \( v_o \) is the speed of the observer (15.3 m/s for the leading ship).
• \( v_s \) is the speed of the source (-15.0 m/s since the trailing ship moves opposite to sound).

Plug these values into the formula to find the new frequency:\[ f' = 1200 \, \text{Hz} \left(\frac{1500 \, \text{m/s} + 15.3 \, \text{m/s}}{1500 \, \text{m/s} - 15.0 \, \text{m/s}}\right) \]The relative motion of both the source and observer affects how they experience the frequency.

Sound in Water

The medium in which sound travels significantly affects how we perceive it. In this problem, sound is moving through water, not air, which changes its speed and the Doppler effect calculation. Sound travels faster in water than in air, primarily because water molecules are more densely packed.

• In water, the speed of sound is around 1500 m/s.
• This speed is much higher compared to air, where sound travels at approximately 343 m/s.

The higher speed in water means that sound waves travel further and faster, affecting how ships at sea might monitor these changes in frequency. Understanding these unique properties of sound in different mediums is essential, especially in scenarios involving sonar signals used often for navigation and detection by ships.