A bus B is moving with a velocity vB in the positive...

A bus $B$ is moving with a velocity $v_{B}$ in the positive $x$-direction along a road as shown in Fig. 9.47. A shooter $S$ is at a distance $l$ from the road. He has a detector which can detect signals only of frequency $1500 \mathrm{~Hz}$. The bus blows horn of frequency $1000 \mathrm{~Hz}$. When the detector detects a signal, the shooter immediately shoots towards the road along $S C$ and the bullet hits the bus. Find the velocity of the bullet if velocity of sound in air is $v=340 \mathrm{~m} / \mathrm{s}$ and $\frac{v_{B}}{v}=\frac{2}{3 \sqrt{3}}$.

Key Concepts

Doppler Effect

This concept describes how the frequency of a wave (in this case, sound) is perceived differently by an observer if the source of the wave is moving relative to the observer. In problems where a moving source emits sound, the frequency heard can be higher or lower than the actual emitted frequency, depending on whether the source is moving towards or away from the observer. Understanding the Doppler effect helps in connecting the emitted frequency and the detected frequency to the relative speed between the source and the observer.

Relative Velocity

Relative velocity is the idea of analyzing the motion of one object in relation to another. In scenarios with multiple moving bodies, such as a moving bus and a bullet fired from a stationary point, it is crucial to account for their individual velocities and directions to determine when and how they meet. This concept allows us to set up equations based on the speeds of the objects as measured in different frames of reference.

Reference Frame Analysis

This concept involves the study of motions from different perspectives or coordinate systems, which can simplify complex problems. By choosing an appropriate reference frame—such as that of the shooter or the bus—one can more easily relate the velocities, directions, and timing of events. A proper frame transformation supports the analysis of how the Doppler shift corresponds to the relative motion of the source and observer.

Geometric Relations in Motion

Geometric analysis in motion problems involves the use of coordinate geometry and trigonometry to relate distances, angles, and directions between moving objects. In situations where the trajectory or line of sight is not aligned with the direction of motion (for example, the shooter’s line to the road), geometry becomes essential for connecting the physical setup with the timing and distance relations needed to determine the collision or interception conditions.

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