An elastic ball is dropped on a long inclined plane. If bounces, hits the plane again, bounces and so on. let us Label the distance between the point of the first and second hit d 12 and the distance between the points of second and the third hit is d 23 . find the ratio of d 12 / d 23 .

Step-by-step solution

If we rotate the coordinate system so that the ramp is horizontal, then the free fall acceleration will have two components one downward a y = - g cos ⁡ θ and one horizontal a x = g sin ⁡ θ. In this frame, the initial velocity will have components given by V y = V 0 cos ⁡ θ & V x = V 0 sin ⁡ θ time for each bounce is given by t b = 2 V y - a y = 2 V 0 g The horizontal displacement Δ x = V x t + 0.5 a x t 2 given these equation d 12 = V 0 sin ⁡ θ 2 V 0 g + 0.5 g sin ⁡ θ 2 V 0 g 2 = 4 V 0 2 sin ⁡ θ g d 13 = V 0 sin ⁡