$A$ plane is inclined at an angle $\alpha = 30^o$ with respect to the horizontal. $A$ particle is projected with a speed $u = 2 \, m/s$ from the base of the plane,making an angle $\theta = 15^o$ with respect to the plane as shown in the figure. The distance from the base at which the particle hits the plane is close to ........ $cm$ (Take $g = 10 \, m/s^2$)

$A$ particle is projected in air at an angle $\beta$ to a surface which itself is inclined at an angle $\alpha$ to the horizontal (figure).
$(a)$ Find an expression for the range on the plane surface (distance on the plane from the point of projection at which the particle will hit the surface).
$(b)$ Find the time of flight.
$(c)$ Find the angle $\beta$ at which the range will be maximum.

$A$ small ball is projected up a smooth inclined plane with an initial speed of $10 \ m/s$ along a direction at $30^{\circ}$ to the bottom edge of the slope. It returns to the edge after $2 \ s$. The ball is in contact with the inclined plane throughout the process. Find the inclination angle $\theta$ of the plane. (in $^{\circ}$)

$A$ particle falling vertically from a height hits a plane surface inclined to the horizontal at an angle $\theta$ with speed $v_0$ and rebounds elastically. Find the distance along the plane where it will hit the second time.

$A$ projectile is launched with a speed of $10\,m/s$ at an angle of $60^{\circ}$ with the horizontal from a sloping surface of inclination $30^{\circ}$. The range $R$ is $........\,m$ (Take $g=10\,m/s^2$).

$A$ man standing on a hill top projects a stone horizontally with speed $v_0$ as shown in the figure. Taking the coordinate system as given in the figure,the coordinates of the point where the stone will hit the hill surface are: