Problem 77 - Physics

Question

Refer to Interactive Solution \(\underline{16.77}\) at for one approach to this type of problem. Two trucks travel at the same speed. They are far apart on adjacent lanes and approach each other essentially head-on. One driver hears the horn of the other truck at a frequency that is 1.14 times the frequency he hears when the trucks are stationary. The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). At what speed is each truck moving?

Step-by-Step Solution

Answer

Each truck is moving at approximately 42 m/s.

Step 1: Doppler Effect Formula

We begin by using the Doppler effect formula for sound. The formula to determine the observed frequency (\(f'\)) is given by:\[f' = f \left( \frac{v + v_o}{v + v_s} \right)\]where:- \(f\) is the actual frequency of the sound.- \(v\) is the speed of sound.- \(v_o\) is the speed of the observer (0 in this case, as the listener is a driver of the approaching truck).- \(v_s\) is the speed of the source of the sound (the other truck).

Step 2: Substitute Stationary Condition

Under stationary conditions, both trucks do not move, and thus the frequency heard matches the source frequency. This simplifies our expression to:\[f' = f\]

Step 3: Apply Condition for Moving Trucks

Now, when the trucks are moving towards each other, the frequency heard is 1.14 times the stationary frequency. According to the condition of the problem:\[f' = 1.14f\]Substituting this into the Doppler effect formula from Step 1:\[1.14f = f \left( \frac{v}{v - v_s} \right)\]

Step 4: Simplify the Equation

We cancel \(f\) on both sides of the equation (assuming \(f eq 0\)):\[1.14 = \frac{v}{v - v_s}\]We will solve this equation to find \(v_s\).

Step 5: Solve for Truck Speed \(v_s\)

Multiply both sides by \(v - v_s\):\[1.14(v - v_s) = v\]Expand and rearrange to isolate \(v_s\):\[1.14v - 1.14v_s = v\]\[v_s(1.14) = 0.14v\]Solving for \(v_s\):\[v_s = \frac{0.14v}{1.14}\]Given that \(v = 343 \,m/s\), substitute to find \(v_s\):\[v_s = \frac{0.14 \times 343}{1.14}\]\[v_s \approx 42 \,m/s\]

Key Concepts

Sound Waves

Sound waves are vibrations that travel through the air or any other medium. They carry energy from one place to another. When an object, like a horn, creates a sound, it sends out waves in all directions. These waves contain compressions and rarefactions of air particles, which reach our ears and are interpreted as sound.

If you imagine dropping a stone into still water, ripples spread out from where the stone landed. Similarly, sound waves spread out from their source. The speed at which these waves travel depends on the medium, such as air, water, or steel. For this problem, we're focusing on air, where sound travels at 343 m/s under standard conditions.

This constant speed of sound in air is important because it helps us determine phenomena like the Doppler Effect, which is crucial when thinking about how people perceive sounds differently based on their movements.

Frequency Shift

The frequency shift is the change in the perceived frequency of a sound compared to its actual frequency. This happens due to the relative motion between the source of sound and the observer. If they're moving towards each other, waves compress, causing a higher frequency sound. If moving apart, the waves stretch, leading to a lower frequency sound.

In our truck problem, the observed frequency is 1.14 times greater than the stationary frequency due to both trucks moving towards each other. This illustrates the concept of frequency shift. This perceived change in pitch happens because the sound waves are "pushed" together as the trucks approach, making them reach the observer's ear more frequently.

Understanding this concept helps explain why sounds like a car horn or a siren change pitch as they pass by.

Relative Motion

Relative motion considers how fast an object moves concerning another. In our scenario, each truck has its speed, but because they're both moving towards each other, their combined effect influences how sound is heard.

Imagine standing still versus running towards a friend who's also running towards you. The speed at which you come together is faster due to your combined speeds. The same principle applies to the trucks and how they affect the sound waves traveling between them. Thus, the relative motion plays a significant role in altering the perceived frequency.

This idea is essential for applying the Doppler Effect, where understanding the motion between source and observer is key to determining the frequency shift.

Speed Calculation

Speed calculation involves using known formulas and physical constants to find unknown speeds. In this problem, we apply the Doppler effect formula to determine the speed of the trucks given the frequency shift and speed of sound.

The equation derived from the Doppler effect tells us how speed changes observed frequency. When rearranging this equation: \[1.14 = \frac{v}{v - v_s}\], we isolate the truck speed \(v_s\) to calculate how fast each truck moves. By inserting the known speed of sound 343 m/s, we solve \(v_s = \frac{0.14 \times 343}{1.14}\) to find that each truck is moving at approximately 42 m/s.

• The speed of sound is 343 m/s helps set the standard.
• Rearranging formulas is crucial for isolating variables.
• Using given frequency ratios helps complete the calculation.

This method shows the step-by-step approach for finding speed using physical principles and basic algebra.