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Question: A train of length \(200\,m\), travelling at \(30\,m{s^{ - 1}}\) overtakes the another train of lengt...

A train of length 200 m200\,m, travelling at 30 ms−130\,m{s^{ - 1}} overtakes the another train of length 300 m300\,m travelling at 20 ms−120\,m{s^{ - 1}}. The time taken by the first train to pass the second train is
(A) 30 s30\,s
(B) 50 s50\,s
(C) 10 s10\,s
(D) 40 s40\,s

Explanation

Solution

The total relative distance to be covered by the overtaking train is equal to the sum of the length of the two trains. And the relative velocity of the first train with respect to the second train is the difference between train velocities. With the help of the speed formula the time taken by the first train to overtake the second train can be obtained.

Formula used:
Moseley’s law of equation is given by,
υ=a(z−b)\sqrt \upsilon = a\left( {z - b} \right)
Where, υ\upsilon is the frequency of the characteristic X-rays, aa and bb are constants and zz is the atomic number.

Complete step by step solution:
When a train of length 200 m200\,m is travelling at 30 ms−130\,m{s^{ - 1}} overtake the another train of length 300 m300\,m travelling at 20 ms−120\,m{s^{ - 1}}. Then the time taken by the first train to the second train is given by the speed formula.
The relative distance covered by the first train to overtake the second train is the sum of the length of two trains.
d=200 m+300 m d=500 m  d = 200\,m + 300\,m \\\ d = 500\,m \\\
The relative velocity of the two trains is the difference in the velocity of two trains.
s=30 ms−1−20 ms−1 s=10 ms−1  s = 30\,m{s^{ - 1}} - 20\,m{s^{ - 1}} \\\ s = 10\,m{s^{ - 1}} \\\
By using the speed, distance and time relation formula,
s=dts = \dfrac{d}{t}
By rearranging the above formula,
t=dst = \dfrac{d}{s}
By substituting the values of distance and velocity in the above equation,
t=500 m10 ms−1t = \dfrac{{500\,m}}{{10\,m{s^{ - 1}}}}
By dividing the above values and cancelling the units,
t=50 st = 50\,s

∴\therefore The time taken by the first train to overtake the second train, t=50 st = 50\,s. Hence, the options (B) is correct.

Note:
When the two trains are travelling in the same direction, then the relative speeds or velocities is the difference between the two speeds or velocities. And the relative distance is the sum of the length of the two trains when the two trains are travelling in the same direction. By the help of the speed formula, the time taken by the first train to pass the second train can be determined.