$A$ particle moving along a straight line covers the first half of the distance with a speed of $3 \, m \, s^{-1}$. The other half of the distance is covered in two equal time intervals with speeds of $4.5 \, m \, s^{-1}$ and $7.5 \, m \, s^{-1}$ respectively. The average speed of the particle during the motion is:

For any arbitrary motion in space,which of the following relations are true?
$(a)$ $v_{\text{average}} = (1/2) (v(t_1) + v(t_2))$
$(b)$ $v_{\text{average}} = [r(t_2) - r(t_1)] / (t_2 - t_1)$
$(c)$ $v(t) = v(0) + at$
$(d)$ $r(t) = r(0) + v(0)t + (1/2)at^2$
$(e)$ $a_{\text{average}} = [v(t_2) - v(t_1)] / (t_2 - t_1)$
(The 'average' stands for the average of the quantity over the time interval $t_1$ to $t_2$.)

$A$ cyclist starts from the centre $O$ of a circular park of radius $1\; km$,reaches the edge $P$ of the park,then cycles along the circumference,and returns to the centre along $QO$ as shown in Figure. If the round trip takes $10 \;min$,what is the
$(a)$ net displacement,
$(b)$ average velocity,and
$(c)$ average speed of the cyclist?

Let $v$ and $a$ denote the velocity and acceleration respectively of a body. Which of the following statements is correct?

$A$ monkey climbs up a slippery pole for $3 \ s$ and subsequently slips for $3 \ s$. Its velocity at time $t$ is given by $v(t) = 2t(3 - t)$ for $0 < t < 3$ and $v(t) = -(t - 3)(6 - t)$ for $3 < t < 6$ in $m/s$. It repeats this cycle until it reaches the height of $20 \ m$.
$(a)$ At what time is its velocity maximum?
$(b)$ At what time is its average velocity maximum?
$(c)$ At what time is its acceleration maximum in magnitude?
$(d)$ How many cycles (counting fractions) are required to reach the top?

$A$ particle moves along the straight line $y=3x+5$. Which coordinate changes at a faster rate?