∑ F = ma
… is a differential equation:
where acceleration a(t), velocity v(t), and displacement s(t) are all vectors and functions of time. This equation is second-order in position because the highest derivative is the second time derivative of position. Combined with the right boundary conditions, s(t) (also called the trajectory: path through space and time) can be determined.
This differential equation can be solved one component, or dimension, at a time. Let us focus on one of these, and call it the x component. The equations for y and z can be found exactly the same way.
If the graph of a(t) signifying acceleration in the x direction is constant
then the graph of v(t), the velocity in the x direction, is a straight line with slope a0
and the graph of x(t), the position along the x axis, is a parabola
It is also possible for the acceleration, or either of the initial velocity or initial position, to be negative. Thus the displacement/projectile motion formula is derived.