Particles P and Q move with constant speeds 2 m s⁻¹ and 4 m s⁻¹ along mutually perpendicular straight lines toward intersection O. At t=0 they are 12 m and 19 m from O. Find the time in seconds when they are nearest. Also calculate shortest distance in the solution.
Physics · Kinematics part-3 Relative motion · Relative motion in 2-D and shortest distance problem
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Question

Particles P and Q move with constant speeds 2 m s⁻¹ and 4 m s⁻¹ along mutually perpendicular straight lines toward intersection O. At t=0 they are 12 m and 19 m from O. Find the time in seconds when they are nearest. Also calculate shortest distance in the solution.

Physics · Kinematics part-3 Relative motion · Relative motion in 2-D and shortest distance problem

G1 Physics · Accepted Answer

D²=(12-2t)²+(19-4t)². Setting d(D²)/dt=0 gives t=5 s. At t=5 s, positions relative to O are 2 m and -1 m, so shortest separation=√5≈2.24 m. Numeric answer=5.