91Ó°ÊÓ

Constant acceleration refers to a situation where an object's rate of change of velocity remains the same over time. This is a key concept in this exercise. The speedboat starts from rest, meaning its initial velocity is zero. It then accelerates at a constant rate of \(3.00 \, \text{m/s}^2\). This uniform increase in velocity allows us to use the kinematic equations, specifically:

• \(v = u + at\), which calculates the final velocity \(v\) based on initial velocity \(u\), acceleration \(a\), and time \(t\).
• Alternatively, \(v^2 = u^2 + 2as\), used here to determine the velocity based on displacement \(s\).

In this problem, we utilize \(v^2 = u^2 + 2as\) because we have the displacement and need to find the velocity reached by that point. With constant acceleration, these equations become powerful tools for predicting future motion, aiding not just in academic exercises but also in real-world applications.

The concept of frequency change is crucial in understanding how motion impacts sound perception. In this exercise, a siren emits a constant frequency of \(755 \, \text{Hz}\). However, a person on a boat moving away from the source perceives a different frequency due to the Doppler Effect.

The Doppler Effect causes the frequency of the sound waves to alter, depending on the relative motion of the source and observer. If the observer moves away from the source, the perceived frequency decreases and vice versa. Here, the boat accelerates away from the dock, causing an apparent increase in the wavelength received and thereby a shift in frequency.

Using the formula:\[f' = \left( \frac{v_{\text{sound}} + v_{\text{observer}}}{v_{\text{sound}}} \right) f\],we determine the new perceived frequency \(f'\). This shift is why moving objects and observers perceive notable changes in pitch—an intriguing real-world example of wave interactions!

Understanding the speed of sound is essential to compute changes in frequency due to motion. The speed at which sound travels through air depends on temperature and is calculated through the equation:\[v_{\text{sound}} = 331.4 + 0.6 \times T\],where \(T\) is the temperature in degrees Celsius.

For this problem, with an air temperature of \(20 \, ^\circ\text{C}\), the speed of sound is \(343.4 \, \text{m/s}\). This plays a vital role in how sound frequencies are perceived as it affects the propagation of sound waves. Calculating this accurately is crucial in practical applications, from acoustics to meteorology.

The fact that the speed of sound varies with temperature is why sound travels faster on a warm day compared to a cold day. Hence, both scientists and engineers must account for such environmental variables when working with sound in various fields.

Motion along a straight line refers to linear motion where an object moves in a single dimension. In this exercise, the speedboat's path directly impacts how sound is perceived by someone on the boat.

Starting from rest, the boat accelerates uniformly, allowing us to track its displacement, velocity, and time using straightforward kinematic equations. This helps simplify problems involving complex motion by reducing them to cases of linear path calculation.

Such linear motion ensures that the concepts like acceleration and displacement are easily quantifiable since only one spatial direction needs consideration. Understanding this type of motion is foundational in physics as it forms the basis for studying more complex trajectories later on.