Analytical mechanics for students of physics and engineering . MOTION 83 Therefore v = Vx- + y- = k Vx- + if= ka. Thus the particle describes a circle with a constant speed ka. The direc-tion of the velocity at any instant is given by the relation tan 6 = I The components vr and vp may be obtained at once by remembering, (1)that the radius vector is constant: , f = 0, (2) that it is always normalto the path: , rdd = ds. Therefore dr n v* = Tt = °»at and dd dsTdt=dt=V ka. 82. Velocity of a Particle Relative to Another Particle in Motion.—Consider the motion of a particle Pi, Fig. 52, wi
Analytical mechanics for students of physics and engineering . MOTION 83 Therefore v = Vx- + y- = k Vx- + if= ka. Thus the particle describes a circle with a constant speed ka. The direc-tion of the velocity at any instant is given by the relation tan 6 = I The components vr and vp may be obtained at once by remembering, (1)that the radius vector is constant: , f = 0, (2) that it is always normalto the path: , rdd = ds. Therefore dr n v* = Tt = °»at and dd dsTdt=dt=V ka. 82. Velocity of a Particle Relative to Another Particle in Motion.—Consider the motion of a particle Pi, Fig. 52, with Y. o x. Fia. 52. respect to a particle P2, when both arc in motion relativeto the system of axes XOY. Let the system of axes A/M have /, for it- origin andmove with its axes parallel to those of the system let (x1} yx) and >x:. //v be the positions, and v, andv2 the velocities of 1\ and Ps with respeel to XOY. Thru 84 ANALYTICAL MECHANICS if (x, y) denotes the position and v the velocity of Pi withrespect to XP-iY, we get x = Xi — x2, y = yi - yi- Differentiating the last two equations with respect to thetime Therefore x = Xi - x2, \i = i/i - i/2. v = x + y= (xi + yi= Vi - v2. ,X2+ \T2 (V) Equation (V) states that the velocity of a particle withrespect to another particle is obtained by subtracting thevelocity of the first from that of the second. ILLUSTRATIVE EXAMPLE. Two particles move in the circumference of a circle with constantspeeds of v and 2 v. Find their relative velocities. Lei the slower one be chosen as the reference particle, and let the angleP2OP1, Fig. 53
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