$A$ car starts from rest and accelerates at $5 \, m/s^{2}$. At $t=4 \, s$,a ball is dropped out of a window by a person sitting in the car. What is the velocity and acceleration of the ball at $t=6 \, s$? (Take $g = 10 \, m/s^{2}$)

$A$ particle is moving in the $xy$-plane in a circular path with its center at the origin. If at an instant the position of the particle is given by $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$,then the velocity of the particle is along .......

At a given instant of time,the position vector of a particle moving in a circle with a velocity $\vec{v} = 3 \hat{i} - 4 \hat{j} + 5 \hat{k}$ is $\vec{r} = \hat{i} + 9 \hat{j} - 8 \hat{k}$. Its angular velocity $\vec{\omega}$ at that time is:

The position vector of a particle is given as $\vec{r} = (t^2 - 8t + 12)\hat{i} + t^2\hat{j}$. The time after which the velocity vector and acceleration vector become perpendicular to each other is equal to ........ $sec$.

$A$ 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 $g = 9.8\,m/s^2$)

Fill in the blanks:
$(a)$ If $\overrightarrow A \cdot \overrightarrow B = AB$,then the angle between $\overrightarrow A$ and $\overrightarrow B$ is ............
$(b)$ The velocity of a projectile at its maximum height is ............ (Take the angle of projection as $\theta$).
$(c)$ The projection of $\widehat i - 2\widehat j + 4\widehat k$ on the $y$-axis is ............