Two cars $A$ and $B$ are racing along straight line. Car $A$ is leading such that their relative velocity is directly proportional to the distance between the two cars. When the lead of car $A$ is $l_1=10 \mathrm{m}$, its running $10 \mathrm{m} {\mathrm{s}}^{-1}$ faster than car $B$. If the time car $A$ will take to increase its lead to $l_2=20 \mathrm{m}$ from car $B$ is $t=\left({\log }_{e}n\right) \sec$, then find $n$.

A large heavy box is sliding without friction down a smooth plane of inclination $\theta$. From a point $P$ on the bottom of the box, a particle is thrown inside the box. The initial speed of the particle with respect to the box is $u$ and the direction of projection makes an angle $\mathrm{\alpha }$ with the bottom as shown in the figure. 

(a) Find the distance along the bottom of the box between the point of projection $P$ and the point $Q$ where the particle lands (assume that the particle does not hit any other surface of the box. Neglect air resistance).

(b) If the horizontal displacement of the particle as seen by an observer on the ground is zero. Find the speed of the box with respect to the ground at the instant when the particle was projected.

An insect moves in a circular path of radius $R$. If it rotates through an angle $\theta$, find its displacement, $\overrightarrow{s}=\mathrm{\Delta }\overrightarrow{r}$, where, $\mathrm{\Delta }\overrightarrow{r}$ is the change in position vector.