At time $t=0$, a particle leaves the origin and moves in the positive direction of the $X$-axis. If the velocity of the particle varies as $v(t)=v_0(1-t/t_0)$, where $|v_0|=10 \ m/s$ and $t_0=10 \ s$, then the distance covered by the particle during the first $20 \ s$ is: (in $m$)

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

$A$ car is moving along a straight line,say $OP$ in the given figure. It moves from $O$ to $P$ in $18\; s$ and returns from $P$ to $Q$ in $6.0\; s$. What are the average velocity and average speed of the car in going from $O$ to $P$ and back to $Q?$

The engine of a motorcycle can produce a maximum acceleration $5 \, m/s^2$. Its brakes can produce a maximum retardation $10 \, m/s^2$. What is the minimum time in which it can cover a distance of $1.5 \, km$?

Two masses of $3.4 \text{ kg}$ and $2.5 \text{ kg}$ are accelerated from an initial speed of $5 \text{ m/s}$ and $12 \text{ m/s}$, respectively. The distances traversed by the masses in the $5^{th}$ second are $104 \text{ m}$ and $129 \text{ m}$, respectively. The ratio of their momenta after $10 \text{ s}$ is $\frac{x}{8}$. The value of $x$ is . . . . . . .

If a car covers $2/5^{th}$ of the total distance with speed $v_1$ and $3/5^{th}$ of the total distance with speed $v_2$,then the average speed is: