1. A particle of mass m = 1 is subject to the force FOE) : (4, 2t, 4) at time t. It has an initial velocity of O and an initial position of {1, 2, ). a) Find the position vector 77(t) of the particle. b) Which direction is the particle traveling at t : 1? What about at t : 1? (Report unit vectors). 0) For 1 g t g 1, when is the particle moving at the slowest Speed? d) How much distance does the particle travel from t = 1 to t = 1? 2. Answer the following questions about a particle of xed mass m. (a) If the particle is only subject to some force Flt), show that the speed |17(t)| is constant if and only if 13" (t) ' 17(t) = 0 for all t. That is, the particle has constant speed throughout its trajectory if and only if at any time t, no component of the force points in the direction of motion. (b) As a consequence of part a, show that if the particle is subject to a constant nonzero force F (that does not depend on time), it cannot maintain a constant speed. (c) If the particle is only subject to some force FOE), show that Flt) is parallel to Flt) for all t if and only if 77(t) X 17(t) is some constant vector 7'73 for all t. Remark: Since F(t) is always perpendicular to 75' = F(t) X 17(t), this shows that if the force is always radial (i.e. parallel to the position F(t)), then the trajectory of the particle is stuck on the plane through the origin with normal vector 11'. 3. The curvature of a parameterized curve r(t) can be computed as k (t ) = IldT/ dill 113 (t) 11 where v(t) = at = dr (t) is the velocity, and T(t) = (t) /| |3(t) | | is the unit tangent vector. Geometrically, larger curvatures correspond to sharper turns, while smaller curvatures correspond to wider turns. (a) Compute the curvature k(t) for the curve r(t) = (Rcos(at), Rsin(at)) for R, a > 0. What is this curve? (b) Compute the curvature k(t) for the curve r(t) = (Rcos(at), Rsin(at), Bt) for R, a, B > 0. What is this curve? (c) In the previous example, how is the curvature affected as you increase S? Explain why this makes sense geometrically in terms of the curve