A motor cyclist travels a certain distance with a uniform speed of $30\, m/s$ and immediately turns back and returns to starting point with a uniform speed $20\, m/s$. Then the average speed of the motor cycle is ……….$m/s$ :-

Explain clearly, with examples, the distinction between 

$(a)$ magnitude of displacement (somettmes called distance) 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].

Give difference between average speed and average velocity.

A point traversed half of the distance with a velocity $v_0$. The remaining part of the distance was covered with velocity $v_1$ for half the time and with velocity $v_2$ for the other half of the time. The mean velocity of the point averaged over the whole time of motion is

A car starts from rest and moves with uniform acceleration $'a'$ on a straight road from time $t = 0$ to $t = T$ . After that, a constant deceleration $'a'$ brings it to rest. In this process the average speed of the car is