An ambulance car is going with a speed of 60 km/h, while a car is trying to go around it in a circle with a speed 50 km/h. If the sound that ambulance car emits has a frequency of 1 kHz, which frequency does the driver of the car hear in Hz when the ambulance car is in the center of the circle it makes, and the car makes an angle θ=30∘ with the direction of the car?
To find out the frequency heard by the driver of the car, we need to consider the Doppler effect. The Doppler effect is an observed shift in the frequency of a wave as the source or observer moves relative to each other.
In this case, we have an ambulance car moving with a speed of 60 km/h emitting a sound wave with a frequency of 1 kHz. The car is trying to go around the ambulance car in a circle with a speed of 50 km/h, making an angle of θ = 30° with the direction of the ambulance car.
First, let's determine the relative velocity between the two cars. Since the car is moving in a circle, we need to consider its tangential velocity, which is given by:
Vt = V * sin(θ)
Substituting the given values, we have:
Vt = 50 km/h * sin(30°)
= (50 km/h) * (1/2)
= 25 km/h
Now, the relative velocity (Vr) between the two cars is the vector sum of their velocities:
Vr = Vambulance - Vcar
Substituting the values, we have:
Vr = (60 km/h) - (25 km/h)
= 35 km/h
Now, let's calculate the frequency heard by the driver of the car using the Doppler effect equation:
f' = f * (Vsound + Vr) / (Vsound + Vs)
Where:
f' = frequency heard by the observer
f = frequency of the source
Vsound = speed of sound (approximately 343 m/s)
Converting the relative velocity to m/s, we have:
Vr = 35 km/h * (1000 m/km) / (3600 s/h)
= 9.72 m/s
Substituting the values into the equation, we have:
f' = 1 kHz * (343 m/s + 9.72 m/s) / (343 m/s + 0 m/s)
= 1 kHz * (352.72 m/s) / (343 m/s)
= 1012.29 Hz
Therefore, the driver of the car hears a frequency of approximately 1012.29 Hz.