Question

A particle moves in the plane with position given by the vector valued function r(t)=cos^3(t)i+sin^3(t)j

MA330 Homework #2 particle moves in the plane with position given by the vector-valued function The curve it generates is called an astrid and is plotted for you below. (a) Find the position att x/4 by evaluating r(x/4). Then draw this vector on the graph (b) Find the velocity vector vt)-r)-.Be sure to apply the power and (e) Find the velocity at t /4 by evaluating v(x/4). Then show that the position and below starting at the origin and determining a position on the curve. while differentiating. velocity are orthogonal at t-/4. Finally, draw the velocity vector on the graph chain rules below tangent to the curve at the appropriate position. (d) Show that the speed is lvO)l-3 co) sin(t), assuming osts/2 You will need to factor and use the identity sin?'o)+)-1. Recall that A Al, for any real number A. How is that relevant here?) (e) It should be intuitively clear that this curve is not smooth. Review the definition of smoothness and explain precisely why r(t) is not smooth on the interval 0,2r), but is smooth on the interval (0,2) In particular, what's happening at the "comers" that results in the non-smoothness? Explain. (0 You can find the arc-length of one quarter of the astrid by integrating its speed from (arc-length)- v3t) sint) dts Evaluate this integral using the u-substitution u - sin(t) What is the length of the entire astrid? Does this agree with the integral3co() sin(e) dt. Why or why not? (% What is the length of the entire astrial Doesthis agqe with the inlegral sint)

Answer 1

r (t) = cos^3 t i + sin^3 t j, 0 lessthanorequalto t lessthanorequalto 2 pi a) r (pi/4) = cos^3 (pi/4)i + sin^3 (pi/4)j = (1/squareroot 2)^3 i + (1/squareroot 2)^3 j = 1/2 squareroot 2 i + (1/squareroot 2)^3 j = 1/2 squareroot 2 i + 1/squareroot 2 j = 0.35 i + 0.35 j b) Velocity vector V(t) = r (t0 = 3 cos^2 t (-sin t) i + 3 sin^2 t (cos t0 j = -3 cos^2 t sin t i + 3 sin^2 t cos t j c) V (pi/4) = -3 cos^2 pi/4 sin pi/4 i + 3 sin^pi/4, cos pi/4 j = -3 times 1/2 middot 1/squareroot 2 i + 3 middot 1/2 middot 1/squareroot 2 j = -3/2 squareroot 2 i + 3/2 squareroot 2 j = -1.06 i + 1.06 j let (x_1, y_1) = (0.35, 0.35) & (x_2, y-2) = (-1.06, 1.06) therefore x_1 x_2 + y_1