Analytical mechanics for students of physics and engineering . When, as in the case of Fig. 48, the particle moves in thejcy-plane, z = 0, therefore The direction of v, in this case, is given by tan0 = H-,x where 0 is the angle v makes with the x-axis. (HI) (IV) ILLUSTRATIVE EXAMPLE. Find the path, the velocity, and the components of the velocity of a,particle which moves so that its position at any instant is given by thefollowing equations: x = at, (a) y=-hgt*. (b) Eliminating t between (a) and (b), we obtain 2a2 for the equation of the path, therefore the path is a parabola, Fig. 49. To fin
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Analytical mechanics for students of physics and engineering . When, as in the case of Fig. 48, the particle moves in thejcy-plane, z = 0, therefore The direction of v, in this case, is given by tan0 = H-, x where 0 is the angle v makes with the x-axis. (HI) (IV) ILLUSTRATIVE EXAMPLE. Find the path, the velocity, and the components of the velocity of a, particle which moves so that its position at any instant is given by thefollowing equations: x = at, (a) y=-hgt*. (b) Eliminating t between (a) and (b), we obtain 2a2 for the equation of the path, therefore the path is a parabola, Fig. 49. To find the component-velocities we differentiate (a) and (b) withrespect to the time. This gives Y x = a, y - -</<• .-. v = /«2 + gt2. Discussion. — The horizontal compo-nent of the velocity IS directed tn the rightand is constant, while the vertical com-ponent is directed downwards and increasesat a constant rate. We will Bee later that these (([nations Il, ;- 49. represent the motion of a body which is projected horizontally from anelevated position.. MOTION 81 PROBLEMS. 1. Find the path and the velocity of a particle which moves bo thatits position at any instant is ijiven by the following pairs of equal ii (a) x = at, V = ht. (b) x = at, y = at- gt°- (c) X = at, y = b cos ut. (d) X = a sin ut, V = bt. (e) x = a sin ut, y = a cos ut. (f) x = a sin ut, y = b sin u& (g) x = aekt, v/ = ae~kt. ve the relation v = Vx2 + y 2 + i2. 81. Radial and Transverse Components of Velocity. — Themagnitude of the velocity along the radius vector is, accord-ing to the results of the preceding section, a) Vr ill The expression for the velocity at right angles to r is ob-tained by considering the motion of the projection of tin-particle along a perpendicular ytor. When the particle movesthrough ds, its projectionmoves through r dd, Fig. 50, therefore the required velocityis rdd dt do -nr— =rd.dt (2)