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Question

Two particles are projected simultaneously from the same point with the same speed but with different angles of projection α and β(α>β). Then,
  1. The line joining the positions of the particle at any subsequent time makes a constant angle  π2+(αβ2) with the horizontal.
  2. The line joining the positions of the particle at any subsequent time makes a constant angle  π2+(α+β2)with the horizontal.
  3. The magnitude of the relative velocity of the first particle with respect to the second is 2usin(α+β2)
  4. The magnitude of the relative velocity of the first particle with respect to the second is 2usin(αβ2)


Solution

The correct options are
B The line joining the positions of the particle at any subsequent time makes a constant angle  π2+(α+β2)with the horizontal.
C The magnitude of the relative velocity of the first particle with respect to the second is 2usin(α+β2)
For the first particle x1=ut cos α,y1=ut sin α12gt2v1x=u cosα;viy=usinαgt;For the second particle, x2=ut sin β,y2=ut sin β12gt2v2x=ucosβ;v2y=usin βgt

Angle made by the line joining the positions of two particles istan θ=y2y1x2x1=u(sin αsinβ)u(cos αcosβ)=cot(α+β2)θ=π2+(α+β2)v12x=u(cosαcosβ) and v12y=u(sinαsinβ). Hence v12=u(cosαcosβ)2+(sinαsinβ)2                =22cosαcosβ+2sinαsinβ=2usin(α+β2)

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