| A driver is caught crossing a red light. The driver claims to the judge the colour she actually saw was green \[({{f}_{G}}=5.6\times {{10}^{14}}Hz)\]and not red\[({{f}_{R}}=4.80\times {{10}^{14}}Hz)\]because of the Doppler effect. |
| The judge accepts this explaination and instead fines her for speeding at the rate of 1Rs. for each kilometer per hour she exceeded the speed limit erf 100 km/h. |
| Find charged will be |
A) Rs.164, 999,999
B) Rs.165, 000, 000
C) Rs.174, 999, 900
D) Rs.164, 999, 900
Correct Answer: D
Solution :
| Longitudinal Doppler effect in light gives, |
| \[f={{f}_{0}}{{\left( \frac{1+v/c}{1-v/c} \right)}^{1/2}}\] |
| where, f = frequency observed, |
| q = frequency emitted by source, |
| v = speed of observer moving towards source |
| and c = speed of light. |
| \[\therefore \] We have, \[v=c\left( \frac{{{f}^{2}}-f_{0}^{2}}{{{f}^{2}}+f_{0}^{2}} \right)\] |
| \[v=3\times {{10}^{8}}\left( \frac{{{\left( 5.6 \right)}^{2}}-{{\left( 4.8 \right)}^{2}}}{{{\left( 5.6 \right)}^{2}}+{{\left( 4.8 \right)}^{2}}} \right)\]\[=4.59\times {{10}^{7}}m/s\]\[=1.65\times {{10}^{3}}km/h\] |
| So, fine \[-\left( 1.65\times {{10}^{8}}-100 \right)\]= 164,999,900 |
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