The position of an object moving along the $x$-axis is given by $x = a + b t^2$,where $a = 8.5 \; m$,$b = 2.5 \; m s^{-2}$,and $t$ is measured in seconds. What is the average velocity between $t = 2.0 \; s$ and $t = 4.0 \; s$ (in $; m s^{-1}$)?

$A$ body covers the first one-third of the distance with a velocity of $20 \, m/s$,the second one-third with a velocity of $30 \, m/s$,and the last one-third with a velocity of $40 \, m/s$. The average velocity is nearly........$m/s$.

Explain clearly, with examples, the distinction between
$(a)$ magnitude of displacement over an interval of time, and the total length of path covered by a particle over the same interval.
$(b)$ magnitude of average velocity over an interval of time, and the average speed over the same interval. [Average speed of a particle over an interval of time is defined as the total path length divided by the time interval].
Show in both $(a)$ and $(b)$ that the second quantity is either greater than or equal to the first. When is the equality sign true? [For simplicity, consider one-dimensional motion only].

$A$ body moves in a straight line with speeds $v_1$ and $v_2$ for distances which are in the ratio $1: 2$. Find the average speed.

$A$ body covers half of its distance with speed $u$ and the other half with a speed $v$,the average speed of the body is

$A$ body has speed $V, 2V$ and $3V$ in the first $1/3$ of distance $S$,second $1/3$ of $S$,and third $1/3$ of $S$ respectively. Its average speed will be: