#54 parts a and b A circular trajectory An object moves clockwise around a circl
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#54 parts a and b
A circular trajectory An object moves clockwise around a circle centered at the origin with radius 5 m beginning the point (0.5) Find a position function r that describes the motion if the object moves with a constant speed, completing 1 lap every 12 s. Find a position function r that describes the motion if it ocean with speed e. A helical trajectory An object moves on the helix (cos t, sin t, t) for t 0. Find a position Junction r that describes the motion if it occurs with a constant speed of 10. Find a position function r that describes the motion if it occurs with speed t. Speed on an ellipse An object moves along an ellipse given by the function r(t) = (a cod t, b sin t). for 0 t 2pi. where a > 0 and b > 0. Find the velocity and speed of the object in terms of a and b.Explanation / Answer
So we won't confuse time t with the parameter t describing the shape of the helix, we'll call that the latter parameter q, so that the helix is <cosq, sinq, q>. We want to find q as a function of t. We know that dr/dt=velocity=10, where r=distance of a point on the helix from a convenient reference, say (0,0,0).
Chain rule says dr/dt = dr/dq.dq/dt; so 10 = dr/dq.dq/dt. dr/dq = d/dq (cos2q + sin2q +q2) = d/dq(1+q2) = 2q/21+q2
So (dq/dt) q/(1+q2) =10, or, qdq/(1+q2) = 10 dt. Put q2=s, so 2qdq=ds, to get ds/(1+s)=20dt, or, upon integrating, 2(1+s) = 20t + c, or (1+q2) = 10t + c where the factor of 2 has been absorbed in c. Let's assume that when t=0, the particle is at q=0, i.e. (1,0,0). (What sort of texbook would give a differential equation and not specify the initial condition?). So c=1, and
q(t)=(10t+1) -1. Thus the position vector r is given by <cosq(t), sinq(t), q(t)> where you substitute the expression (10t+1) -1 in the place of q(t)
(b). I suppose now t represents the time, so velocity = time in some units. dr/dt is now t instead of 10. Repeating the anslysis, qdq/(1+q2) = t dt instead of 10 dt. 2(1+q2) = t2 + c; c=2 if we assume q=0 at t=0.
Inverting, q(t)=[(1/4)(t2 + 2)2-1]. Thus the position vector r is given by <cosq(t), sinq(t), q(t)> where you substitute the expression [(1/4)(t2 + 2)2-1] in the place of q(t).
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