The trajectory of a particle in projectile motion is given by $y = x - \frac{x^2}{80}$. Here, $x$ and $y$ are in meters. For this projectile motion, match the following with $g = 10 \, m/s^2$.

$Column-I$	$Column-II$
$(A)$ Angle of projection	$(p)$ $20 \, m$
$(B)$ Angle of velocity with horizontal after $4 \, s$	$(q)$ $80 \, m$
$(C)$ Maximum height	$(r)$ $45^{\circ}$
$(D)$ Horizontal range	$(s)$ $\tan^{-1}(1/2)$

$A$ particle is performing uniform circular motion. If $\theta$,$\omega$,$\alpha$,and $a$ are its angular displacement,angular velocity,angular acceleration,and centripetal acceleration respectively,then which of the following is '$WRONG$'? ($v$ is its linear velocity)

If $T$ is the total time of flight,$h$ is the maximum height,and $R$ is the horizontal range,how are the $x$ and $y$ coordinates of projectile motion related to time $t$ and range $R$?

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.

$A$ particle is rotating in a circle of radius $1\,m$ with constant speed $4\,m/s$. In time $1\,s$,match the following (in $SI$ units) columns.

Column $I$	Column $II$
$(A)$ Displacement	$(p)$ $8 \sin 2$
$(B)$ Distance	$(q)$ $4$
$(C)$ Average velocity	$(r)$ $2 \sin 2$
$(D)$ Average acceleration	$(s)$ $4 \sin 2$

$A$ particle is performing uniform circular motion with angular momentum $L$. If the frequency of motion is doubled and the kinetic energy is halved,the new angular momentum will be