Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Euler Method for a Mass on a Spring

 The Euler method treats Newton's second law (N2) as a set of "marching orders." Given the "state of an object (its position and velocity) at a particular time, t, it uses N2 to calculate (predict) the state of the object (its position and velocity) at a later time, t + Δt. (a) For a cart on a spring, taking the coordinate to start at 0 at the equilibrium position, write N2 for the cart. (b) Rewrite N2 for the time interval from t to t + Δt and show how to rewrite this form to give Euler's method -- a way of calculating the position and velocity at a time t + Δt given the position and velocity at a time t.
(c) If at t = 0, the position of the object is x(0) = A and the velocity is v(0) = 0, find the position and velocity of the mass as given by the first two steps of the Euler method. ("Two" does not include the initial state.) How good are these answers?

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