# a particle is projected from surface of the inclined plane with speed u at an angle theta with the horizontal.after some time the particle collides elastically with smooth fixed inclined plane for the first time and subsequently moves in vertical direction.starting from projection.find the time taken by the particle to reach maximum height

Dear student,

Let angle of inclined plane from horizontal is $\alpha$.
Time of flight of particle in projectile motion before collision is given by,
${t}_{1}=\frac{2u\mathrm{sin}\left(\theta -\alpha \right)}{g\mathrm{cos}\alpha }$
The particle will have velocity u after collision.
So,time in vertical motion,
${t}_{2}=\frac{u}{g}$
$T={t}_{1}+{t}_{2}=\frac{2u\mathrm{sin}\left(\theta -\alpha \right)}{g\mathrm{cos}\alpha }+\frac{u}{g}$
Regards.

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