I. E. Irodov Solution 1.2

Solution to I. E. Irodov in General Physics and H. C. Verma in Concept of Physics is like a bible for student who are appearing for IIT-JEE, JEE/Main and JEE/Advance, UG NEET, AIIMS or any other Engineering and Medical entrance examination. All the questions in these books are of high level, which requires all basics to applied concept of physics. We here makes your task very easy, we have presented complete solution with detailed explanation step by step. The solution of these books teaches students also teachers in a suitable manner and then tests you with some tricky questions. To answer these questions you need to have thorough understanding of the concepts and this is where most students falter.

Problem: 1.2
A point traversed half a distance with a velocity \({V_0}\). The remaining part of the distance was covered with velocity \({V_1}\) for half the time and with velocity \({V_2}\) for the other half of the time. Find the mean velocity of the point average over the whole time of motion.

Solution: 1.2
Mean velocity is total distance by total time. Let the total distance traveled be \(d\). The time taken to travel half the distance \(\left( {\frac{d}{2}} \right)\) at speed \({V_0}\) is \( = \left( {\frac{d}{{2{V_0}}}} \right)\).
Now another \(\left( {\frac{d}{2}} \right)\) distance remains to be traveled.
Now let the time taken to travel this remaining distance be \(t\).
Then, \(\left( {\frac{t}{{2{V_1}}} + \frac{t}{{2{V_2}}}} \right) = \left( {\frac{d}{2}} \right)\)
This means that, \(t = \left( {\frac{d}{{{V_1} + {V_2}}}} \right)\)

The total time traveled thus is, \(\left( {\frac{d}{{2{V_0}}}} \right) + \left( {\frac{d}{{{V_1} + {V_2}}}} \right)\).
Here the total distance is \(d\).
Thus the mean velocity is = \(\left( {\frac{d}{{\frac{d}{{2{V_0}}} + \frac{d}{{\left( {{V_1} + {V_2}} \right)}}}}} \right) = \frac{{2{V_0}\left( {{V_1} + {V_2}} \right)}}{{2{V_0} + {V_1} + {V_2}}}\)