4.32. A particle falling vertically from a height hits a plane surface inclined to horizontal at an angle θ with speed v0v0 and rebounds elastically. Find the distance along the plane where it will hit the second time.
Step 1: For the motion of the projectile from O to A.
Considering x and y-axes as shown in the diagram.
y = 0, uy = v0cosθay = −gcosθ, t = T
Applying the equation of kinematics
y = uyt+12ayt2⇒ 0= v0cosθT+12(−gcosθ)T2⇒T[v0cosθ−gcosθ,T2]=0T=2v0cosθgcosθ
Hence, T=2v0g
Step 2: Now considering motion along OX.
x=L,ux=v0sinθ,ax=gsinθ,t=T=2v0g
Applying the equation of kinematics,
x=uxt+12axt2⇒ L=v0sinθt+12gsinθt2=(v0sinθ)(T)+12gsinθT2=(v0sinθ)(2v0g)+12gsinθ×(2v0g)2=2v20gsinθ+12gsinθ×4v20g2=2v20g[sinθ+sinθ]⇒ L=4v20gsinθ
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