$A$ bogie of a uniformly moving train is suddenly detached from the train and stops after covering some distance. What is the relation between the distance covered by the bogie and the distance covered by the train in the same time?

$A$ bead is moving in medium $1$ with a uniform speed of $1\, m/s$ for $2.5\, s$. Then it enters into air and falls freely under gravity for $2\, m$. Finally, it enters medium $2$ and immediately moves with uniform speed for $1.5\, s$. The total distance the bead has traveled is.........$m$ $(g = 10\, m/s^2)$: (in $.1$)

$A$ stone is dropped into a lake from a tower $500 \ m$ high. The sound of the splash will be heard by the man approximately after ..... $s$. (Take $g = 10 \ m/s^2$ and speed of sound $v = 340 \ m/s$)

When the velocity of a body is variable,then:

$A$ particle moves along the sides $AB, BC, CD$ of a square of side $25 \, m$ with a constant velocity of $15 \, m \, s^{-1}$. Its average velocity is ........ $m \, s^{-1}$.

$A$ car travels from $A$ to $B$ (without changing direction). During the first part of the journey,its average speed is $V_1$,and for the second part of the journey,its average speed is $V_2$. The ratio of the path length of the second part to the path length of the first part is $\sqrt{\frac{V_2}{V_1}}$. In this case,the average speed of the total path is the ....... of the average speeds of both parts. Choose the correct option.