Question

Problem 4. Given a curve C, the vectors T(t), N(t), and B(t) form a special coordinate system (called an orthonormal reference frame) that lets us discuss velocity and acceleration of a moving object from the perspective of the object itself. (Consider, for example, looking only at the motion of an airplane to study its stability without worrying about its position relative to its starting point.) (a) Use the fact that v uT, where u(t)-|r(t)l is the speed of the particle, along with one of the differentiation rules above (specify which one!) to show (b) Show T'(t) (t)o(t)N(t), where k(t) is the curvature at time t. (Hint: Begin by showing IT (t)]()(t), and use the fact that T (t) and N(t) have the same direction.)

Answer 1

Given that , a curve c, the vector T(t) , N(t) and B (t) from a special co-ordinate system a) the fact that v = VT where v (t) = Vertical-bar r (t) Vertical-bar is the speed of particle along with one of the differentiation rules are a(t) = v(t) t(t) + v(t) t (t) v = vt a (t)) = dv/dt \s\n we can the different ion chain rule is a (t) = dv/dt = d(vt)/dt = vdt/dt + r d(v)/dt Because a (t) = v (t) t (e) + v(t) T (t) b) T (t) = k(t) v (t) n(t) where k (t) is the curvature at time t begin by showing (+) Vertical-bar = k (t) v (t) the fact that t(t) and N(t) is the same direction Because dt/dt = t(t) = k(t v (t) n (t) \s\n the acceleration of a particle is given by a = v t + kv^2 n v(t) the tangential component