Final answer:
To find the speed of the car, we can use the formula for the Doppler effect and set up a system of equations using the given frequencies. Solving the system, we find that there is no valid speed for the car that would result in the given frequencies.None of the above options are correct.
Step-by-step explanation:
In this scenario, we are dealing with the Doppler effect, which is the apparent change in frequency or pitch of a sound or light wave due to relative motion between the source of the wave and the observer. The formula for the Doppler effect is given as:
f' = f * (v + vo) / (v - vs)
Where:
- f' is the perceived frequency
- f is the actual frequency
- v is the speed of sound in air
- vo is the velocity of the observer (in this case, the student)
- vs is the velocity of the source (in this case, the car)
To find the speed of the car, we can set up an equation using the given frequencies:
76 = 65 * (343 + vo) / (343 - vs)
Simplifying this equation, we get:
76 * (343 - vs) = 65 * (343 + vo)
We also know that the speed of sound is given as 343 m/s, so we can substitute this value into the equation:
76 * (343 - vs) = 65 * (343 + vo)
Simplifying further:
26168 - 76vs = 22295 + 65vo
Moving like terms to one side of the equation, we get:
76vs + 65vo = 3882
Now we need another equation to solve for the speed of the car. Since the frequency decreases after the car passes, we use a negative value for the velocity of the source:
-65 = 76 * (343 + vo) / (343 - (-vs))
Simplifying this equation, we get:
-65 * (343 - (-vs)) = 76 * (343 + vo)
Substituting the value of 343 for the speed of sound:
-65 * (343 + vs) = 76 * (343 + vo)
Simplifying further:
-22295 - 65vs = 26168 + 76vo
Moving like terms to one side of the equation, we get:
65vs + 76vo = -48463
Now we have a system of two equations:
76vs + 65vo = 3882
65vs + 76vo = -48463
To solve this system, we can use any method of solving simultaneous equations, such as substitution or elimination. Using the elimination method, we can multiply the first equation by 65 and the second equation by 76 to eliminate the coefficient of the variable 'vs'. This gives us:
4940vs + 4225vo = 252330
4940vs + 5856vo = -368233
Subtracting the first equation from the second equation, we get:
1626vo = -620563
Solving for 'vo', we find:
vo = -381.42 m/s
Since the velocity of the observer cannot be negative, we discard this solution. Therefore, there is no valid speed for the car that would result in the given frequencies. The correct answer is none of the provided options (a), (b), (c), or (d).