A policeman with a very good ear and a good understanding of the Doppler effect stands on the shoulder of a freeway assisting a crew in a 40-mph work zone. He notices a car approaching that is honking its horn. As the car gets closer, the policeman hears the sound of the horn as a distinct G3 tone (196 Hz). The instant the car passes by, he hears the sound as a distinct D3 tone (147 Hz). He immediately jumps on his motorcycle, stops the car, and gives the motorist a speeding ticket. How fast was this car going? (Assume that the speed of sound is 340 m/s.)

196=f1/(340-Vcar)

147=f(1/(340+vcar)

dividing the first equation by the second

196/147=(340+vcar)/(340-Vcar)

4/3 (340-Vcar)=340+Vcar

1/3 340=7/3 Vcar

vcar=340/7=you do it... m/s That is almost 109mph

To solve this problem, we need to use the Doppler effect formula, which relates the observed frequency of a sound to the motion of the source and the observer.

The formula for the observed frequency of a sound when the source is moving is given by:

fobs = (v + vo) / (v + vs) * f

Where:
fobs = observed frequency
f = actual frequency of the sound
v = speed of sound
vo = velocity of the observer (positive if moving towards the source, negative if moving away from the source)
vs = velocity of the source (positive if moving away from the observer, negative if moving towards the observer)

Let's assume the speed of sound, v, is 340 m/s.

Given:
fobs when the car is approaching = G3 = 196 Hz
fobs when the car passes by = D3 = 147 Hz

We need to find vs, the velocity of the source (the car).

When the car is approaching, vo (velocity of the observer) is positive.
When the car passes by, vo becomes negative as the policeman is now moving towards the car on his motorcycle.

Using the formula, we can set up two equations:

196 = (340 + vo) / (340 - vs) * f ---(1)
147 = (340 - vo) / (340 + vs) * f ---(2)

Now, let's solve these equations to find vs, the velocity of the source (the car).

First, let's solve equation (1) for vo:

196(340 - vs) = (340 + vo)f ---(3)

Next, let's solve equation (2) for vo:

147(340 + vs) = (340 - vo)f ---(4)

By combining equations (3) and (4), we can eliminate vo:

196(340 - vs) = 147(340 + vs)

Simplifying this equation:

66,464 - 196vs = 50,580 + 147vs

343vs = 15,884

vs ≈ 46.26 m/s

Finally, we can convert the velocity of the source (car) from m/s to mph:

46.26 m/s * 2.23694 = 103.4 mph

Therefore, the car was traveling at a speed of approximately 103.4 mph.