Displacement, velocity and acceleration

Displacement, velocity, and acceleration are the kinematic quantities that describe the motion of an object. The displacement defines the position of the object at any given time, the velocity is the variation of position in time (i.e. the time derivative of the displacement), and the acceleration is the variation of velocity in time (i.e. the time derivative of the velocity).

For a mass oscillating on a vertical spring according to simple harmonic motion, the displacement as a function of time is a sinusoidal function. As a result, the velocity and acceleration also change in a sinusoidal manner. Figure 1 shows the vertical (and only) component of displacement, x(t), velocity, v(t), and acceleration, a(t) for a mass that is initially displaced at a distance A from its equilibrium position (here set as x=0).

Figure 1: Displacement, velocity and acceleration for a mass on a vertical spring.

Once the mass is left free to move at x=A, it starts moving symmetrically around the equilibrium position. In particular:

• At the equilibrium position (x=0 in the Fig.1), the total force and therefore the acceleration of the mass are both zero (Fig. 1, bottom), while the velocity is at its maximum (Fig. 1, middle). Maximum velocity means also maximum kinetic energy. This is why the mass cannot be found at rest at the equilibrium position once it is displaced from it (in ideal conditions).

• At the turning points (the points of maximum displacement, x=A, and x=-A) the mass stops moving before inverting its motion and the velocity is zero. Here the acceleration (and the total restoring force) is maximum.

The displacement, velocity, and acceleration all oscillate with the same period, T. This is independent of the amplitude, A: no matter how far from the equilibrium the mass is initially displaced, the period will stay the same (in the linear regime).