Question

A train has each compartment of mass $m$ is accelerating uniformly. At the instant they cross a certain mile maker, the velocity of the engine is $v_1$ and the velocity of the last compartment is $v_2$. Find the momentum of the compartment in the middle when it passes the same mile marker assuming that the compartments are of point sizes.

• A. $m\sqrt{\frac{V_1^2-V_2^2}{2}}$
• B. $m\sqrt{\frac{V_1^2{V}_2^2}{2}}$
• C. $m\times (\frac{V_1^2-V_2^2}{2}$)
• D. $m\times (\frac{V_1^2+V_2^2}{2}$)

Answer 1

The correct option is B $m\sqrt{\frac{V_1^2{V}_2^2}{2}}$


For the portion AC,
$V^2-V_1^2=2ax$
For the portion CB
$V_2^2-V^2=2ax$
$\Rightarrow V=\sqrt{\frac{V_1^2+V_2^2}{2}}$
$\Rightarrow p=mv=m\sqrt{\frac{V_1^2+V_2^2}{2}}$