A differential equation for the velocity of a falling mass subjected to air resistance proportional to the square of the instantaneous velocity is m(dv/dt)=mg-kv2 Where ? > 0 is a constant of proportionality. The positive direction is downward. (a) Solve the equation subject to the initial condition ?(0) = ?0 . (b) Use the solution in part (a) to determine the terminal velocity of the mass. (c) If the distance s, measured from the point where the mass was released above ground, is related to velocity v by ds/dt=v(t) , find an explicit expression for s(t) if s(0)=0

A model for the position x(t) of an Olympic jump skier says that the acceleration a(t) = x" is proportional to the square of the velocity v(t) = x'(t). Note this model comes from taking into account the friction of air against the skier, and we are greatly simplifying the problem by not taking into account gravity. In this problem, units of distance are in meters and units of 1. time are in seconds. (a) Give an ODE in terms of v(t) only (meaning: no x or a) that corresponds to this model, and find its general solution v. (b) Recalling that our model comes from taking friction into account, which will slow down the skier, say whether you expect your proportionality constant from la to be positive or negative. (c) Use your solution v(t) from la to find x(t). (d) Use the initial conditions' x(0) = 0 and v(0) = 100 to find the unique solution for x(t), the position of the skier. (e) Assume for simplicity the following information: the skiers take 8 seconds before reaching the ground after…

Suppose we throw a stone up into the air. It's velocity in meters/sec after t seconds is given by v (t) = 30 – 10t (a) Find the displacement of the stone from t seconds. = 0 seconds to t 30 (b) Find the total distance the stone travels from t = 0 seconds to t = 30 seconds.

(1) A particle has an initial displacement of s = 30 metres when t = 0. The velocity of the particle is v= 6t² - 216t+ 1944 m/s (t > 0). (a) Find the displacement of the particle at time t. (b) Find the acceleration of the particle at time t. (c) What is the acceleration and displacement when v = 0?