A body travels a distance s in t seconds.

Question

A body travels a distance $s$ in $t$ seconds. It starts from rest and ends at rest. In the first part of the journey it moves with constant acceleration $f$ and in the second part with constant retardation $r$. The value of $t$ is given by:

Accepted Answer

Let the common speed at the junction be $v$, with $AB=x$ (acceleration phase) and $BC=y$ (retardation phase), $s=x+y$.

From $v^2=2fx$: $x=\dfrac{v^2}{2f}$, and $v=ft_1 \implies t_1=\dfrac vf$.

From $v^2=2ry$: $y=\dfrac{v^2}{2r}$, and $v=rt_2 \implies t_2=\dfrac vr$.

$$x+y=\frac{v^2}2\left(\frac1f+\frac1r\right)=s,\qquad t=t_1+t_2=v\left(\frac1f+\frac1r\right)$$

$$\frac{t^2}{2s}=\frac{v^2\left(\frac1f+\frac1r\right)^2}{2\cdot\frac{v^2}2\left(\frac1f+\frac1r\right)}=\frac1f+\frac1r \implies t=\sqrt{2s\left(\frac1f+\frac1r\right)}$$

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