Question

A train moves in straight line with a uniform acceleration. If x and y be the velocities with which the front and rear end of the train respectively cross a fixed pole then the velocity with which the middle of the train crosses the pole is:

• A. $\frac{x^2+y^2}{2}$
• B. $\frac{2xy}{x+y}$
• C. $\sqrt{\frac{1}{2}(x^2+y^2)}$
• D. $\sqrt{\frac{1}{2}(y^2-x^2)}$

Answer 1

Answer: (C)

$\sqrt{\frac{1}{2}(x^2+y^2)}$

Let the total length of train is ${}^{\prime }{L}^{\prime }$

Since $X=$front end velocity, and $Y=$rear end velocity

Uniform acceleration of train$=am/s^2$

From equation of motion(when whole train just cross the pole)

$Y^2=X^2+2aL$

$aL=(Y^2-X^2)/2$

Suppose V' be velocity when half the train corsses the pole

$V^{\prime 2}=X^2+2aL/2$

$V^{\prime 2}=X^2+aL$

$V^{\prime 2}=X^2+(Y^2-X^2)/2$

$V^{\prime 2}=\frac{X^2+Y^2}{2}$