If a particle moving on the Euclidean line traverses distance $1$ in time $1$ starting and ending at rest, then at some time $t \in [0, 1]$, the absolute value of its acceleration should be at least $4$.
Please give me a hint to proceed
2026-02-23 03:57:01.1771819021
Prove by Taylor expansion or mean value theorem
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I suppose you are given the functional form $v$ of the speed. Scalar acceleration $a$ is the derivative $v'$ of the speed. Then by the mean value theorem, for every pair of $\xi, \eta \in (0,1)$ there is some $c \in [\xi,\eta]$ such that $v(\xi) - v(\eta) = a(c)(\xi - \eta)$. Can you proceed?