If $h_{1}, h_{2}$ and $h_{3}$ are the displacements of object along the incline plane between $1 s t - 2 n d, 2 n d - 3 r d$ and $3 r d - 4 t h$ impact, then $h_{1} : h_{2} : h_{3}$ .

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Solution

R.E.F image
Suppose the object coincides at a velocity
V in the $1 s t$ impact.
Considering t challenges a projective motion the
velocity before the $2^{n d}$ concision is V as were.
Suppose, the coefficient of restitution $= e$
Where, $e < 1$ (for non - elastic concision)
The velocity after $2^{n d}$ concision is ev
and $3^{r d}$ collision is $e^{2} v$
Let the angle of inclined power be $θ$
Vertical component of velocity after
$1 s t$ concision is $v c o s θ$
$2^{n d}$ concision is $e v c o s θ$
$3^{r d}$ concision is $e^{2} v c o s θ$
$\frac{1}{2} m (v c o s θ)^{2} = m g h_{1}$
$h_{1} = \frac{(v c o s θ)^{2}}{2 g}$
$h_{2} = \frac{(e v c o s θ)^{2}}{2 g}$
$h_{3} = \frac{(e^{2} v c o s θ)^{2}}{2 g}$
$∴ h_{1} : h_{2} : h_{3} = 1 : e^{2} : e^{4}$

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If h1,h2 and h3 are the displacements of object along the incline plane between 1st−2nd,2nd−3rd and 3rd−4th impact, then h1:h2:h3.

If $h_{1}, h_{2}$ and $h_{3}$ are the displacements of object along the incline plane between $1 s t - 2 n d, 2 n d - 3 r d$ and $3 r d - 4 t h$ impact, then $h_{1} : h_{2} : h_{3}$ .