$A$ 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 $^{\circ}$)

If the maximum range down an inclined plane is three times the maximum range up the same inclined plane,find the angle of inclination in degrees.

$A$ projectile is fired with a velocity $v$ at a right angle to a slope which is inclined at an angle $\theta$ with the horizontal. The expression for the range $R$ along the incline is

$A$ 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 projected with speed $u$ at an angle of $60^o$ with the horizontal from the foot of an inclined plane. If the projectile hits the inclined plane horizontally,the range on the inclined plane will be:

$A$ 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.