A particle has two velocities of equal magnitude inclined to each other at an angle θ. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then θ is

90°

120°

45°

60°

Solution

120°

tan space straight theta over 4 space equals space fraction numerator begin display style straight u over 2 end style space sin space straight theta over denominator straight u space plus begin display style straight u over 2 end style space cos space straight theta end fraction rightwards double arrow space sin space straight theta over 4 space plus space 1 half sin space straight theta over 4 space cos space straight theta space equals space 1 half space sin space straight theta space cos space straight theta over 4 therefore space 2 space sin space straight theta over 4 space equals space sin space fraction numerator 3 straight theta over denominator 4 end fraction space equals space 3 space sin straight theta over 4 minus 4 space sin cubed space straight theta over 4 therefore space sin squared space straight theta over 4 space equals space 1 fourth rightwards double arrow space straight theta over 4 space equals space 30 to the power of straight o straight theta space equals space 120 to the power of straight o

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