A point moves in the $x y$-plane according to the following equation, $x=a \sin \omega t, y=a(1-\cos \omega t)$, where $a$ and $\omega$ are positive constants. Find the angle between the point's velocity and acceleration vectors.

A particle of mass $m=1 \mathrm{~kg}$ moves in the $x y$-plane. The force on it at time $t$ is $F(t)=[2 \sin (\alpha t) \hat{\mathbf{i}}+3 \cos (\alpha t) \hat{\mathbf{j}}] \mathrm{N}$, where $\alpha=1 \mathrm{~s}^{-1}$. At time $t=0$, the particle is at rest at the origin. Calculate the magnitude of its position vector $\mathbf{r}$ (in m ) and velcoity vector $\mathbf{v}$ (in m/s) at time $t=\frac{\pi}{2} \mathrm{~s}$.

A particle aimed at a target, projected with an angle $15^{\circ}$ with the horizontal is short of the target by 10 m . If projected with an angle of $45^{\circ}$ is away from the target by 10 m , then the angle of projection to hit the target is

A person moves 30 m North and then 20 m towards East and finally $30 \sqrt{2} \mathrm{~m}$ in South-West direction. The displacement of the person from the origin will be