PHYSICS-BASED MATH 138 Winter 2016Assignment 5Topics: Parametric curves, Differential equations.Due: 2pm Friday, February 12.1. A curve parameterized in polar coordinates is given by the vector function~γ(θ) = (r(θ) cosθ, r(θ) sinθ), withθ1≤θ≤θ2.(a) Show that the length of the curve is,length(~γ) =θ2Zθ1sr2+drdθ2dθ.(b) Find the length of the polar curver(θ) = 1 + cosθ, with0≤θ≤2π.2. When a raindrop falls, it increases in size and so its mass at timetis afunction oft,m(t).The rate of growth of the mass iskm(t)for somepositive constantk.Applying Newton’s Law of Motion to the raindropresults in,ddt(mv) =gm,wherevis the velocity of the raindrop (directed downward) andgis theacceleration due to gravity. Theterminal velocityvterof the raindrop is,vter= limt→∞v(t).Find an expression for the terminal velocity in terms ofgandk.3. Solve the following differential equations. Note you may not be able to findan explicit expression fory(x)in some cases.