$A$ stone thrown with velocity $u$ at angles $\theta$ and $(90^{\circ}-\theta)$ with the horizontal reaches maximum heights $H_1$ and $H_2$ respectively. Its horizontal range is

$A$ body projected along a frictionless inclined plane of angle of inclination $30^{\circ}$ covers a distance of $40\, m$ along the plane. If the body is projected with the same speed at an angle of $30^{\circ}$ with the ground,it will have a range of: (Take $g = 10\, m/s^2$)

Consider a circle of radius $R = 42\ cm$. An insect crawls with a uniform speed of $v = 1.3\ cm/s$ along the chord $AB$, then along the circular arc $BCD$, and finally along the chord $DA$ to return to point $A$. The time taken by the insect to crawl from $A$ back to $A$ is closest to ......... $\sec$.

$A$ train is moving along a straight line with a constant acceleration '$a$'. $A$ boy standing in the train throws a ball forward with a speed of $10 \ m/s$,at an angle of $60^{\circ}$ to the horizontal. The boy has to move forward by $1.15 \ m$ inside the train to catch the ball back at the initial height. The acceleration of the train,in $m/s^2$,is

Motion in two dimensions in a plane can be studied by expressing position,velocity,and acceleration as vectors in Cartesian coordinates $\vec{A} = A_{x} \hat{i} + A_{y} \hat{j}$,where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ directions,respectively,and $A_{x}$ and $A_{y}$ are corresponding components of $\vec{A}$. Motion can also be studied by expressing vectors in circular polar coordinates as $\vec{A} = A_{r} \hat{r} + A_{\theta} \hat{\theta}$,where $\hat{r} = \cos \theta \hat{i} + \sin \theta \hat{j}$ and $\hat{\theta} = -\sin \theta \hat{i} + \cos \theta \hat{j}$ are unit vectors along the directions in which $r$ and $\theta$ are increasing.
$(a)$ Express $\hat{i}$ and $\hat{j}$ in terms of $\hat{r}$ and $\hat{\theta}$.
$(b)$ Show that both $\hat{r}$ and $\hat{\theta}$ are unit vectors and are perpendicular to each other.
$(c)$ Show that $\frac{d}{dt}(\hat{r}) = \omega \hat{\theta}$,where $\omega = \frac{d\theta}{dt}$ and $\frac{d}{dt}(\hat{\theta}) = -\omega \hat{r}$.
$(d)$ For a particle moving along a spiral given by $\vec{r} = a\theta \hat{r}$,where $a = 1$ (unit),find the dimensions of $a$.
$(e)$ Find velocity and acceleration in polar vector representation for a particle moving along the spiral described in $(d)$ above.

The ratio of minimum kinetic energies of two projectiles of same mass is $4: 1$ and the ratio of maximum heights attained by them is $4: 1$. Then the ratio of their ranges is . . . . . . (in $: 1$)