08 Time, Speed and Distance

When two bodies are moving in same direction with speeds \({S_1}\) and \({S_2}\) respectively, their relative sped is the difference of their speeds. 

Relative Speed = \({S_1} - {S_2}\). 

When two bodies are moving in opposite direction with speeds \({S_1}\) and \({S_2}\) respectively, their relative speed is the sum of their speeds. 

Relative Speed = \({S_1} + {S_2}\)     

Example 9: Two persons A and B started from the same place and in the same direction, after 2 hours distance between them is observed is 3.6 km, if the speed of the faster man is 2 m/s, find the speed of the other man.

Solution: Suppose the speed of the other man is \(x\), then their relative speed is \(2 - x\) and the distance between them after 2 hours is 3.6 km, thus

\(2{\rm{ }} \times {\rm{ }}60{\rm{ }} \times {\rm{ }}60{\rm{ }} \times {\rm{ }}\left( {2{\rm{ }}--x} \right){\rm{ }} = {\rm{ }}3600 \Rightarrow 2{\rm{ }}--x = {\rm{ }}0.5\)

or \(x = 1.5\) m/s

CROSSING OF TWO TRAINS

If 2 trains are running in the same directions, their speeds are \({V_1}\) & \({V_2}\) and their length are \({L_1}\) & \({L_2}\)

Time taken by faster train to over take the slower train 

T = \(\frac{{{L_1} + L2}}{{{V_1} - {V_2}}}\)

In opposite direction time taken to cross each other = \(\frac{{{L_1} + {L_2}}}{{V1 + V2}}\)

Time taken by a train of length \(L1\), to cross the platform of length \(L\)

\(=\frac{{L + {L_1}}}{V}\), where \(V\) is the speed of the train.

Example 10: A train 110 m in length travels at 60 km/h. How much time does the train take in passing a man walking at 6 km/hr against the train? 

Solution: Relative speeds of the train and the man =  (60 + 6) = 66 km/h = \(\frac{{66 \times 5}}{{18}}m/s\)

Distance = 110 m 

 Therefore, time taken in passing the men 

= \(\frac{{110 \times 8}}{{66 \times 5}} = 6\) sec.

Example 11: A cyclist is moving at the speed of 18 Km/hr. Because of the fog he can see only up to 100 m, if a train whose length is 200 mts. overtaking the cyclist and visible to cyclist only for 15 sec. Find speed of train ?

Solution: Total distance traveled by the train w.r.t. the cyclist is 300 mts., hence

\(\frac{{300}}{{v - 5}}\) = 15 sec or \(v  = 25\) m/sec = 90 km/hr

Example 12: Two trains 137 meters and 163 meters in lengths are running towards each other on parallel lines, one at the rate of 42 kmps and another at 48 kmph. In what time will they be clear of each other from the moment the meet? 

Solution:  Relative speed of the trains = \((42 + 48)\)

\(=  90 kmph = 90 \times \frac{5}{{18}}= 25\) m/sec.

Time taken by the trains to pass each other

\( = \left( {\frac{{137 + 163}}{{25}}} \right) = \frac{{300}}{{25}}\)= 12 sec.