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# Linear Systems with Addition or Subtraction

## Solve systems using elimination of one variable

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High Pitch, Low Pitch

When an ambulance passes by with its siren blaring, why does the siren's pitch sound higher when it is approaching you than when it is moving away? Originally proposed by Austrian physicist Christian Doppler, the Doppler effect is a phenomenon that explains this apparent change in pitch.

#### The Doppler Effect

Sound travels through the air in waves. Each sound has a frequency, the number of waves that pass a given point each second, measured in hertz (Hz). The pitch of a sound is your ears' interpretation of the sound's frequency; the higher the frequency, the higher the pitch you perceive.

The sound of an ambulance siren has a frequency that is subject to the Doppler effect because there is relative motion between the source (the ambulance) and the observer (you). This means that the original frequency emanated by the source, the actual pitch of the siren, is not what the observer experiences; you hear a different pitch.

The observed frequency is higher than the original when the distance between the source and the observer is decreasing (whether it be because the source is moving or because the observer is). In other words, the observed frequency is higher when the relative motion between the source and the observer is toward each other. On the other hand, the observed frequency is lower than the original when the distance between the source and the observer is increasing, when the relative motion is away from each other.

#### Explore More

Below is the Doppler equation. If a source emanates a wave with frequency f0\begin{align*}f_0\end{align*}, it will appear as wave with frequency f\begin{align*}f\end{align*} to an observer, as defined by this equation:

f=(c+vrc+vs)f0\begin{align*}f=\left(\frac{c+v_r}{c+v_s}\right)f_0\end{align*}

where c\begin{align*}c\end{align*} is the velocity of waves in the medium, vr\begin{align*}v_r\end{align*} is the velocity of the receiver or observer relative to the medium (positive if te receiver is moving away from the source and negative if moving towards the source), and vs\begin{align*}v_s\end{align*} is the velocity of the source relative to the medium (positive if moving away and negative if moving towards).

When an ambulance approaches you, you record the pitch of its siren at 3166 Hz. When it moves away, the pitch is 2850 Hz. If you're stationary and the speed of the ambulance is constant, how fast is it moving? Take the velocity of sound to be 343 m/s.

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