Question

An engine of the train moving with uniform acceleration passes an electric pole with velocity $u$ and the last compartment with velocity$v$. the middle point of the train passes past the same pole with the velocity of

• A. $\frac{u+v}{2}$
• B. $\sqrt{\frac{u^2+v^2}{2}}$
• C. $\sqrt{\frac{u+v}{2}}$
• D. $\sqrt{\frac{v^2-u^2}{2}}$

Answer 1

The correct option is B $\sqrt{\frac{u^2+v^2}{2}}$

Let the length of the train be $2s$.

Apply the equation of motion: $v^2-u^2=2as$;

$v^2-u^2=2a\left(2s\right)$;

$v^2-u^2=4as$;

$=a=\frac{v^2-u^2}{4s}......Eq:01$

According to question, for half train, the length is $\frac{2s}{2}=s$ and

velocity is V;

Again apply the equation of motion: $v^2-u^2=2as$;

$V^2-u^2=2a\left(s\right)$

Put the value of acceleration (a) from Eq:01 in above equation;

$V^2=u^2+2\left(\frac{v^2-u^2}{4s}\right)\left(s\right)$;

$V^2=\frac{4u^2+2v^2-2u^2}{4}$

$V^2=\frac{u^2+v^2}{2}$

$V=\sqrt{\frac{u^2+v^2}{2}}$

Correct option is (B).