Velocity of an object in rectilinear motion is given as a function of time by $v = t - t^3$,where $v$ is in $m/s$ and $t$ is in seconds. Its average velocity over the time interval from $t = 0 \ s$ to $t = 2 \ s$ is ........ $m/s$.

$A$ person walks along a straight road from his house to a market $2.5 \,km$ away with a speed of $5 \,km/h$ and instantly turns back and reaches his house with a speed of $7.5 \,km/h$. The average speed of the person during the time interval $0$ to $50 \,min$ is (in $m/s$)

$A$ person travelling on a straight line moves with a uniform velocity $v_1$ for a distance $x$ and with a uniform velocity $v_2$ for the next $\frac{3}{2} x$ distance. The average velocity in this motion is $\frac{50}{7} \ m/s$. If $v_1$ is $5 \ m/s$,then $v_2 = $ . . . . . . $m/s$.

$A$ car moves from $X$ to $Y$ with a uniform speed $v_1$ and returns from $Y$ to $X$ with a uniform speed $v_2$. The average speed for this round trip is

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].

Explain average velocity and average speed.