Two particles of masses m and 2m start moving simultaneously from the same point on a smooth horizontal circular track in opposite directions. Both particles move with the same constant speed v . What is the magnitude of the total linear momentum of the two-particle system at the instant when each particle has covered an angular distance of 45^ from the starting point?

Step-by-step solution

Let the particles start from the point (R, 0) on a circular track of radius R centered at the origin. The first particle (mass m ) moves counter-clockwise. When it covers an angular distance of 45^ , its position vector makes an angle of +45^ with the x-axis. Its velocity vector is tangential to the circular path, making an angle of 45^ + 90^ = 135^ with the positive x-axis. The second particle (mass 2m ) moves clockwise. When it covers an angular distance of 45^ , its position vector makes an angle of -45^ with th