Recovering Motion from Initial Data

Integrating acceleration gives velocity; integrating velocity gives position. These are two separate integrations, so determine a separate constant each time. The condition may be given at any time, not only at the start. If acceleration is a(t) and v(t_0)=v_0, then v(t)=v_0+ _t_0ta(u), du. If s(t_0)=s_0, then s(t)=s_0+ _t_0tv(u), du. The letter u is only the integration variable. For the illustrative constant acceleration a=2, with v(1)=3 and s(1)=-4, integration gives v=2t+C_1, hence C_1=1. Then s=t2+t+C_2, hence C_2=-6. Substitution at t=1 checks both conditions. At a change of motion model, carry the position and velocity at the end of the first stage into the second stage. If the second stage uses a new clock T=t-t_0, its initial conditions occur at T=0. A new formula does not imply a return to the origin or to rest.