$A$ projectile is thrown upward at an angle $60^{\circ}$ with the horizontal. The speed of the projectile is $20 \text{ m/s}$ when its direction of motion is $45^{\circ}$ with the horizontal. The initial speed of the projectile is . . . . . . in $\text{m/s}$.

$A$ particle is moving with a constant speed of $\sqrt{2} \, m/s$ on a circular path of radius $10 \, cm$. Find the magnitude of the average velocity when it has covered $\frac{3}{4}$ of the circular path.

At a given instant of time,two particles have position vectors $4 \hat{i} + 4 \hat{j} + 57 \hat{k} \ m$ and $2 \hat{i} + 2 \hat{j} + 5 \hat{k} \ m$ respectively. If the velocity of the first particle is $0.4 \hat{i} \ ms^{-1}$,what is the velocity of the second particle in $ms^{-1}$ if they collide after $10 \ s$?

$A$ particle moves with a constant speed of $5 \, ms^{-1}$ as shown in the figure. What is the change in velocity during half a revolution in $ms^{-1}$?

Fill in the blanks:
$(a)$ For vectors $\vec{A} = 3\hat{i} + 2\hat{j}$ and $\vec{B} = \hat{i} + \hat{j} - 2\hat{k}$,the $y$-component of the vector $(\vec{A} - \vec{B})$ is .....
$(b)$ The formula for the total time of flight of a projectile projected vertically upwards with an initial velocity $u$ is ..........
$(c)$ The formula for the centripetal acceleration of a particle moving in a circular path of radius $R$ with a constant angular velocity $\omega$ is ......
$(d)$ The component of any vector is always .........

$A$ projectile crosses two walls of equal height $H$ symmetrically as shown. If the horizontal distance between the two walls is $d = 120\, m$ and the times at which the projectile crosses the walls are $t_1 = 2\, s$ and $t_2 = 6\, s$,then the range of the projectile is ........ $m$.