Speed cameras make use of the Doppler effect - Leaving Cert Physics - Question c - 2020

The Doppler effect can be explained through visualisation:

• When a source of sound moves towards a stationary observer, the sound waves are compressed. This results in a higher frequency, which means the observer hears a higher pitch. Diagrams may show non-concentric circles moving closer together as the source approaches the observer.

• Conversely, if the source moves away from the observer, the sound waves are stretched, leading to a lower frequency or pitch. In this case, the circles would be seen spreading out.

In summary:

• Source moves towards observer → frequency is higher → wavelength is shorter.
• Source moves away from observer → frequency is lower → wavelength is longer.

Given that the observed frequency ( f') is 20% greater than the frequency at rest ( f), we can write:

$f' = 1.2f$.

Using the Doppler effect formula for the speed of the source:

$f' = f \frac{v + v_o}{v - v_s}$,

where:

• (v) = speed of sound in air (340 m/s),
• (v_o = 0) (stationary observer), and
• (v_s) = speed of the source.

Since the observer is stationary, the formula simplifies to:

$f' = f \frac{v}{v - v_s}$.

Substituting for (f'):

$1.2f = f \frac{340}{340 - v_s}$.

Canceling (f):

$1.2 = \frac{340}{340 - v_s}$.

Cross-multiplying gives:

$1.2(340 - v_s) = 340$,

which simplifies to:

$408 - 1.2v_s = 340$

Thus,

$1.2v_s = 408 - 340$

Resulting in:

$1.2v_s = 68\text{ m/s}$

So,

$v_s = \frac{68}{1.2} = 56.7\text{ m/s}.$

The red-shift observed in galaxies indicates that they are moving away from us. This phenomenon provides evidence for the expansion of the universe. The greater the red-shift, the faster the galaxy is receding, suggesting that the universe is not static but rather is continuously expanding over time.