$A$ ball is dropped from a height of $49\,m$. The wind is blowing horizontally. Due to wind,a constant horizontal acceleration is provided to the ball. Choose the correct statement$(s)$. (Take $g = 9.8\,m/s^2$)

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.

$A$ man standing on the roof of a house of height $h$ throws one particle vertically downwards and another particle horizontally with the same velocity $u$. The ratio of their velocities when they reach the earth's surface will be

If $\alpha$ is angular acceleration,$\omega$ is angular velocity,and $a$ is the centripetal acceleration,then which of the following is true?

$A$ particle of mass $m$ is projected at a velocity $u$, making an angle $\theta$ with the horizontal ($x$-axis). If the angle of projection $\theta$ is varied keeping all other parameters same, then the magnitude of angular momentum $(L)$ at its maximum height about the point of projection varies with $\theta$ as,

At any time $t$,the coordinates of a moving particle are $x = at^2$ and $y = bt^2$. The speed of the particle is