Uniform Acceleration and Retardation

Uniform Acceleration and Retardation

The third CAPS motion law. The follower accelerates uniformly over the first half of the rise, then decelerates uniformly over the second half. Because the acceleration is constant rather than instantaneous, the force on the follower is constant too — the smoothest of the three for high-speed work.

The displacement is parabolic. Under constant acceleration displacement grows with the square of time, so the divisions follow the ratio 1 : 3 : 5 for the accelerating half, and 5 : 3 : 1 for the decelerating half.

The construction. Divide the total rise in the proportion 1 : 3 : 5 : 5 : 3 : 1 for six divisions. A convenient way is the equal-division construction from Grade 10: step off 1 + 3 + 5 + 5 + 3 + 1 = 18 equal parts on an angled line and transfer by parallels.

Worked example. A 36 mm rise over six divisions gives 36 ÷ 18 = 2 mm per unit, so the steps are 2, 6, 10, 10, 6, 2 mm. The middle steps are five times the first — that is the parabola.

Core checkpoint: The odd-number ratio 1 : 3 : 5 is the signature of constant acceleration. Equal steps would be uniform motion instead.