Problem 1. Find the lengths of the curves (a) r(t) : (221,252, t3> for 0 g t s 1. (b) r(t) = (act, c") for 0 S t S 1. Problem 2. (a) Find the length of the curve r(t) : for 0 S t S g (b) Find the curvature of r(t) = (t, t2, t3) at (1,1, 1). Problem 3. Find the position vectors r(t) of a particle with acceleration function a(t) = . Problem 4. For this problem. vectors are in the xyplane, so 2 = 0. A projectile is red With muzzle speed 150 m/s and angle of elevation 450 from a position 10 In above ground level. (a) Denote by r(t) the trajectory of the projectile. by V(t) its velocity, and by a(t) its acceleration. Then a(t) : for all t and r(0) : (0, 10, 0). What is the value of v(0)? (b) Where does the projectile hit the ground (that is, when is y(t) : 0), and with What speed? Problem 5. Let r(t) be a curve with unit speed. Are the velocity v(t) : r'(t) and acceleration a(t) : r\"(t) perpendicular for all values if? Motivate your answeri if yes, explain Why this is the case; if no, provide an example Where this does not happen