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# Two particles are simultaneously prjected in the same vertical plane fomthe same point with velocities u and v at angleg alpha and beta with horizontal. find the time that elapses when theit velocities are parallel?

9 years ago

velocities at time t are:

ucos(alpha)i+(usin(alpha)-gt)j

vcos(beta)i+(vsin(beta)-gt)j

velocities are parallel when:

(usin(alpha)-gt)/ucos(alpha)=(vsin(beta)-gt)/vcos(beta)

9 years ago

If at any instant of time they have parallel velocity, then the angle mad by them with the horizontal should be equal.

If the angles are equal then the tan of the angle made by them wiith the horizontal should also be equal.

At any instant the tan of the angle for the  particlse is vertical component /horizontal componet of velocity = ( v sin B - gt )/v cos B  and  ( u sin A - gt )/u cos A

= ( v sin B - gt )/v cos B =  ( u sin A - gt )/u cos A (A used for alpha and B for Beta)

= uv sinB cosA - ugt cos A = uv sinA cosB - vgt cosB

= uv(sinB cosA - sinA cos B)= gt(ucosA - VcosB)

= uvsin(B-A)/g(ucosA-vcosB) = t