Axes and Components. In the following equations for the motion of a particle with constant acceleration it is assumed that an x,y coordinate system has been chosen and that the particle has the same acceleration everywhere, with constant x and y components ax and ay.
Sign Changes. In the diagram, the direction of the acceleration vector is such that both the x and y components of the vector are positive. In another situation, the direction of the vector may be different, so that one or both of the x and y components may be negative. (Or the direction of the vector may be as shown, but the directions of the x and y axes may have been chosen differently, again causing one or both components of the vector to be negative.) The equations below apply to all such situations. No signs in these equations need to be changed.
2D vs. 1D. The following equations are formulated for the 2D case, motion in two dimensions. However, they also apply to 1D motion, motion along a straight line. In this case, it will often be convenient to choose either the x or y axis to be along the line of motion. Assuming the motion is along the x axis, you will need only the equations for the x components. All y components in any of the equations will be zero. Similarly, if the motion is restricted to the y-axis, just use the y equations and ignore the x equations.
Acceleration, Velocity, and Position vs. Time:
| x components | y components | |
| Acceleration | ax = const | ay = const |
| Velocity | vx(t) = vx(0) + axt | vy(t) = vy(0) + ayt |
| Position | x(t) = x(0) + vx(0)t + (ax/2)t2 | y(t) = y(0) + vy(0)t + (ay/2) t2 |
Speed vs. Displacement:
Δv2 = v2(t) - v2(0) = 2axΔx + 2ayΔy
For 1D motion, say, along the x axis, the speed vs. displacement relation reduces to
Δv2 = v2(t) - v2(0) = 2axΔx = 2ax[x(t) - x(0)] .
On Page 2, all of these equations are written down in vector notation, which makes them look much more compact.