A Course in Applied Mathematics, Vol. 1 and 2 by Derek F. Lawden

By Derek F. Lawden

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Find the velocity with which it must be projected from the point C, where A C = 9GB, in order that it may just come to rest at the middle point 0 of AB. ) Let x be the distance of P from 0 at any instant. The resultant force acting upon the particle in the direction of OP is then mp. mp. a - x + constant. 2 • 2 _ = 2--2 • 36 A C O U RS E I N A P P L I E D MAT H EM A T I C S [CH. We require that when x = 0, then v = 0. , 4/Lx• a(a• - x2) • (i) x + a = 9(a - x) x = ta. For this value of x, equation (i) shows that v2 = 641-' 9a the negative sign being taken, since the velocity of projection from C is clearly towards 0.

S. system, the tractive force due to the engine is Pfv lbals. The equation of motion of the car during any phase of its motion when no change of gear is made is accordingly I·6 I5y'I0/2 n= where 2·5, 30 M dvdt M is the car's mass in lb. = A I5 (nfu) �vi I + (nvfu) 3' Separating the variables we obtain � dt = [(E) -4 v-i + (S) � v�] av. l,. , and t0, t1 are the times at the beginning and end of the phase respectively. In absolute units, A = X X ft lbal sec-1, u = ft sec-1, = lb. ) ��v'. , we find for the time spent in bottom gear 3·76 sec.

If (x, y) A particle moves along the curve are its coordinates at time t, show that (%, y) are the coordinates of the corresponding point on the hodograph with respect to parallel axes. Deduce that if x and y are quadratic functions of t, the hodograph is a straight line. I2. A particle P moves in a plane and has polar coordinates (r, 6) . Its velocity always makes an angle with the radial coordinate r. Show that its track is a straight line or a circle. In the latter case, show that, if the particle's acceleration is always in the radial direction and = 0 when t = 0, then t is proportional to - sin cos e e e e e.

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