4.A fire truck emitting a 450 Hz signal passes by a stationary detector. The difference in frequency measured by the detector is 58 Hz. If the speed of sound is 345m/s how fast is the fire truck moving?

Here's how to solve this Doppler effect problem:<br /><br />**1. Understand the Doppler Effect**<br /><br />The Doppler effect describes the change in frequency of a wave (like sound) as the source and the observer move relative to each other. When the source and observer are moving closer, the observed frequency is higher than the emitted frequency. When they are moving apart, the observed frequency is lower.<br /><br />**2. Identify the Given Information**<br /><br />* f_s (source frequency) = 450 Hz<br />* Δf (change in frequency) = 58 Hz<br />* v (speed of sound) = 345 m/s<br /><br />**3. Determine the Observed Frequencies**<br /><br />Since the fire truck passes the detector, there are two observed frequencies:<br /><br />* f_approach (frequency as the truck approaches): f_approach = f_s + Δf/2 = 450 Hz + 58 Hz/2 = 479 Hz<br />* f_recede (frequency as the truck recedes): f_recede = f_s - Δf/2 = 450 Hz - 58 Hz/2 = 421 Hz<br /><br />**4. Apply the Doppler Effect Formula**<br /><br />The Doppler effect formula for a moving source and a stationary observer is:<br /><br />* **Approaching:** f_approach = f_s * (v / (v - v_s))<br />* **Receding:** f_recede = f_s * (v / (v + v_s))<br /><br />where:<br /><br />* v_s is the speed of the source (the fire truck).<br /><br />**5. Solve for the Speed of the Fire Truck (v_s)**<br /><br />We can use either the approaching or receding formula. Let's use the approaching formula:<br /><br />f_approach = f_s * (v / (v - v_s))<br /><br />479 Hz = 450 Hz * (345 m/s / (345 m/s - v_s))<br /><br />Now, solve for v_s:<br /><br />1. Divide both sides by 450 Hz: 1.0644 = 345 m/s / (345 m/s - v_s)<br />2. Multiply both sides by (345 m/s - v_s): 367.288 m/s - 1.0644v_s = 345 m/s<br />3. Subtract 367.288 m/s from both sides: -1.0644v_s = -22.288 m/s<br />4. Divide both sides by -1.0644: v_s ≈ 20.94 m/s<br /><br />**Answer:**<br /><br />The fire truck is moving approximately 20.94 m/s.<br />