A buzzing fly moves in a helical path given by the equation r(t) = ib sin wt + j
ID: 1999379 • Letter: A
Question
A buzzing fly moves in a helical path given by the equation r(t) = ib sin wt + jb cos wt + kct^2 Show that the magnitude of the acceleration of the fly is constant, provided b, w, and c are constant. A bee goes out from its hive in a spiral path given in plane polar coordinates by r = be^kt theta = ct where b, k, and c are positive constants. Show that the angle between the velocity vector and the acceleration vector remains constant as the bee moves outward. Work Problem 1.18 using cylindrical coordinates where R = b, phi = wt, and z = ct^2. The position of a particle as a function ol time is given by r(t) = i(1 - e^-kt) + je^kt where k is a positive constant. Find the velocity and acceleration of the particle. Sketch trajectory.Explanation / Answer
r(t) = (b Sinwt) i^ + (b Coswt) j^ + (c t2) k^
taking derivative both sides
v(t) = (bw Sinwt) i^ + (bw Coswt) j^ + (2c t) k^
taking derivative both sides
a(t) = (bw2 Sinwt) i^ + (bw2 Coswt) j^ + (2c) k^
Magnitude = sqrt((bw2 Sinwt)2 + (bw2 Coswt)2 + (2c)2 ) = sqrt((bw2)2 + (bw2)2 + (2c)2 )
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