The 2 ends of train ,moving with a constant acceleration,pass a certain with velocity u and v .Show that velocity with which the middle point of train passes the same point is root of(usquare +v square)/2

Asked by ajaysekharjs | 27th Aug, 2018, 11:54: AM

Expert Answer:

Using the equation of kinematics for velocities, acceleration and displacement,
 
begin mathsize 12px style straight v squared equals straight u squared minus 2 as
Where space
apostrophe straight v apostrophe space is space the space final space velocity space of space the space train comma
apostrophe straight u apostrophe space is space the space initial space velocity space of space the space train comma
apostrophe straight a apostrophe space is space the space acceleration space and space apostrophe straight s apostrophe space is space the space displacement space made space by space it.

Therefore comma
as space equals space fraction numerator straight v squared minus straight u squared over denominator 2 end fraction space... left parenthesis straight a right parenthesis
end style
At the velocity at the mid-point of the train is vm 
and that can be found using the same kinematical equation.
 
At the midpoint, the 'u' is zero and the displacement made is,
sm , such that sm = s/2.
 
begin mathsize 12px style straight v subscript straight m squared equals 2 as subscript straight m
straight v subscript straight m squared equals 2 straight a cross times straight s over 2
straight v subscript straight m squared equals as space... space left parenthesis straight b right parenthesis
end style
Using equation (a) and (b)
begin mathsize 12px style straight v subscript straight m squared space equals space fraction numerator straight v squared minus straight u squared over denominator 2 end fraction
straight v subscript straight m space equals square root of fraction numerator straight v squared minus straight u squared over denominator 2 end fraction end root end style
Hence it is proved that the velocity of a train at its midpoint is the square root of half of the difference of the squares of their final and initial velocity.

Answered by Abhijeet Mishra | 28th Aug, 2018, 02:28: PM