The displacement $x$ of a particle depends on time $t$ as $x = \alpha t^2 - \beta t^3$.

Which of the following curves possibly represent one-dimensional motion of a particle? Choose the correct answer from the options given below.

$A$ stone is dropped into a lake from a tower $500 \ m$ high. The sound of the splash will be heard by the man approximately after ..... $s$. (Take $g = 10 \ m/s^2$ and speed of sound $v = 340 \ m/s$)

$A$ body of mass $0.40 \;kg$ moving initially with a constant speed of $10 \;m s^{-1}$ to the north is subject to a constant force of $8.0 \;N$ directed towards the south for $30 \;s$. Take the instant the force is applied to be $t=0$,the position of the body at that time to be $x=0$,and predict its position at $t=-5 \;s, 25 \;s, 100 \;s$.

$A$ ball is thrown vertically upward from a location and returns to the same location. Which of the following statements are correct?
$(a)$ Distance travelled by the ball can be zero.
$(b)$ Displacement of the ball is zero.
$(c)$ Average velocity of the ball is zero.
$(d)$ Acceleration of the ball is zero.

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).