# Which is the correct formula for change in displacement?

I am trying to find distance between two bodies A and B ($s$) as a function of time ($t$) given acceleration ($a$) as a function of distance. My approach is to model the situation with a differential equation that i then can solve.

An object A is released, with initial velocity $0$, $d$ meters above a static body B. A accelerates towards (or away from) B with the acceleration $a(s)$.

Is it correct to express the displacement as $$s(t+\Delta t)=s(t)+v(t)\Delta t$$ or as $$s(t+\Delta t)=s(t)+v(t)\Delta t+\frac{a(s(t))\Delta t^2}{2}~?$$

Note: Here i let $\Delta t$ tend towards $0$.

• Thank you to @Philip Wood. In other news, why am i getting downvoted? – Ola Apr 26 '17 at 10:26
• I did not downvote, but the problem is not formulated clearly. For instance it is not clear (1) which of the two bodies has initial velocity $v_0$, (2) why B would accelerate towards A or away from it, (3) what you are trying to calculate, (4) what you call displacement here? Also, are you familiar with differential calculus, because that's how a physicist would normally write these equations? – user1583209 Apr 26 '17 at 10:41

The first equation is a first order approximation which will be useful only if $\Delta t$ is very small (unless the acceleration is zero throughout the interval, in which case it is exactly correct). The second equation is a second order approximation, which will be useful (that is not too far out!) over larger time intervals, $\Delta t$, but is exactly correct if the acceleration is constant.