$A$ horse rider covers half the distance with $5\,m/s$ speed. The remaining part of the distance was travelled with speed $10\,m/s$ for half the time and with speed $15\,m/s$ for the other half of the time. The mean speed of the rider averaged over the whole time of motion is $\frac{x}{7}\,m/s$. The value of $x$ is

Fill in the blanks:
$(a)$ Average velocity ....... average speed.
$(b)$ $A$ particle moves in a straight line with an initial velocity $v_0$ and constant acceleration $a$. The formula for the distance covered in the $n^{th}$ second is ............ .
$(c)$ When two objects are moving in the same direction with velocities $v_A$ and $v_B$,the formula for the velocity of $A$ relative to $B$ is .......... .

$A$ 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?

The equation of motion of a body is $\frac{dv}{dt} = -4v + 8$,where $v$ is the velocity in $m/s$ and $t$ is the time in seconds. The initial velocity of the particle was zero. Then,

Which of the following statements are correct?
$(a)$ Average speed of a particle in a given time interval is never less than the magnitude of average velocity.
$(b)$ It is possible to have a situation in which $|d\vec{v}/dt| \neq 0$ but $d/dt|\vec{v}| = 0$.
$(c)$ The average velocity of a particle is zero in a time interval. It is possible that the instantaneous velocity is never zero in the interval.
$(d)$ The average velocity of a particle moving on a straight line is zero in a time interval. It is possible that the velocity is never zero in the interval. (Infinite accelerations are not allowed).

What is reaction time? On what does the reaction time depend?