# Accumulations are simple

### Really simple

Velocity is often defined as the rate of change of displacement, and acceleration as the rate of change of velocity, and there comes a time along the learning pathway when this conceptually charged formalisation, with its infinitesimals, is an essential part of the physicist's toolkit. But perhaps not when starting out. Here integration, as simple repetitive addition, seems much more straightforward than subtraction and division, so differentiation. 'Acceleration accumulates velocity' ( and the more demotic version, 'acceleration tells velocity how to change' is more approachable still, but remains operationally precise. 'Acceleration accumulates velocity' is intrinsically a more straightforward formulation than 'acceleration is the rate of change of velocity', as assayed by the number and subtlety of mathematical operations required.

Indeed, numerical methods for solving sets of equations use just such repetitive addition to predict the velocity from a given acceleration, and the displacement from a given velocity. Of course, such predictions are risky: here the velocity as reported by the GPS-enabled device does not guarantee that I'll end up in the sea and miss Edinburgh, or that I'll travel directly from the current location, shown by the cross-hairs, to Edinburgh.

Either approach must be sensitive to the real situation: in the case of modelling with either rates or with accumulations, the situation determines the interval used for calculations.

This consideration becomes much more persuasive when considering vectors as a part of the didactic mix, as adding vectors can be made rather straightforward by treating them as arrows and doing the addition graphically.