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Chapter 9: Problem 247

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 \(\frac{v_{B}}{v}=\frac{2}{3 \sqrt{3}}\).

Short Answer

Expert verified
The velocity of the bullet is \(v_{bullet} = 340\sqrt{3}\, m/s\).

Step by step solution

01

Write the Doppler effect formula

We can use the Doppler effect formula for sound waves to relate the observed frequency (\(f_o\)), the source frequency (\(f_s\)), the speed of sound (\(v\)), and the relative velocities of the source and observer (shooter): \[ f_o = \frac{v + v_o}{v - v_s}f_s \] In our case, the shooter is stationary, so \(v_o = 0\). Also, since the bus is moving in the positive x-direction, we have \(v_s = -v_B\).
02

Plug in the given values and solve for velocity

We're given that \(f_o = 1500\, Hz, f_s = 1000\, Hz, v = 340\, m/s\), and \(\frac{v_B}{v} = \frac{2}{3\sqrt{3}}\). Substituting these values in the Doppler effect formula: \[ 1500 = \frac{340 + 0}{340 - (-v_B)}1000 \] Now, we have: \[ \frac{3}{2} = \frac{340}{340 + v_B} \]
03

Solve for \(v_B\)

We can now solve for \(v_B\): \[ v_B = 340\left(\frac{2}{3} - 1\right) = -\frac{2}{3} \cdot 340 \]
04

Compute the velocity of the bullet

Considering a right triangle with distance \(l\) from the shooter to the road and the horizontal velocity of the bus, we can compute the distance traveled by the bus while the sound travels to the shooter as: \[ l = \frac{-v_B}{v} \cdot \frac{l}{\sqrt{3}} \] Therefore, the velocity of the bullet must be: \[ v_{bullet} = \frac{l \cdot \sqrt{3}}{l} \cdot 340 = 340\sqrt{3} \] The velocity of the bullet is \(v_{bullet} = 340\sqrt{3}\, m/s\).

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Most popular questions from this chapter

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