A particle starts from rest and accelerates along a straight line at $$a_1$$ for some time and then at $$a_2$$ for an equal time covering a distance $$d_1$$. If the acceleration for the first interval were $$a_2$$ and for the second $$a_1$$ the distance travelled is $$d_2$$. How do the distances $$d_1$$ and $$d_2$$ compare? Support your answer either qualitatively or quantitatively.

PLACEHOLDER ANSWER - the answer is a comparison. With each interval of length $$t$$, $$d_1 = \frac{1}{2}a_1 t^2 + (a_1 t)t + \frac{1}{2}a_2 t^2 = \frac{3}{2}a_1 t^2 + \frac{1}{2}a_2 t^2$$, and similarly $$d_2 = \frac{3}{2}a_2 t^2 + \frac{1}{2}a_1 t^2$$. So $$d_1 - d_2 = (a_1 - a_2)t^2$$. The distance is greater when the larger acceleration comes first, since the particle then moves faster for longer; $$d_1 > d_2$$ if $$a_1 > a_2$$, and they are equal if $$a_1 = a_2$$.