Two integrations and two conditions

Acceleration gives v= a,dt, followed by s= v,dt. Each indefinite integration introduces its own constant. Why: acceleration determines changes in velocity but cannot tell us the starting velocity; even after velocity is known, a starting position is still needed. Method: integrate acceleration; use the velocity condition at its stated time; integrate the resulting complete velocity; use the position condition at its stated time. Differentiate both results as a check. If a=2t, v(1)=3 and s(1)=4, then v=t2+C gives C=2. Next s=t3/3+2t+D gives D=5/3. Check: do not set C equal to the velocity given at a nonzero time. Do not use one constant for both integrations; constant acceleration can coexist with changing velocity.