Parabolic Motion or Motion with Constant Acceleration and Retardation:

The technique of constructing displacement diagram is as:

 (a) Divide displacement interval into acceleration & deceleration part. If and retardation and acceleration are equivalent than this shall be θ_o /2 or θr/2 for acceleration and θ_o/2 or θr/2 for declaration. If acceleration is twice the retardation, then, it shall be θ_o/3 or θr/3 for acceleration as well as 2θ/3 or 2θr/3 for retardation. Likewise, the lift is divided. Divide acceleration & retardation parts into equivalent parts say n = 4.

(b) Spilt lift portions in n² or say 16 parts.

(c) Project 1 to primary ordinate, 2² as 4 to second ordinate, 3² as 9 to third ordinate, 4²  as 16 to fourth ordinate to get point on displacement diagram.

 (d) Repeat the process mentioned in (c) for other part to obtain remaining part of displacement diagram.

The diagram is illustrated in Figures (a) and (b).

As motion is parabolic

Let  y = C θ²

here C is constant of proportionality.

Let  y = L/2  when θ= θ_o/2 (for equivalent acceleration and retardation)

Putting  for C

Thus,

and acceleration

Maximum velocity

Maximum acceleration

For return stroke the following expression present

The, velocity, displacement and acceleration diagrams are shown in Figure.