A subway train travels over a distance $s$ in $t$ seconds. It starts from rest (zero velocity) and ends at rest. In the first part of its journey, it moves with constant acceleration $f$ and in the second part with constant deceleration (negative deceleration) $r$. Show that $$s = \frac{\frac{frt^2}{f + r}}{2}$$
I tried doing this multiple times but to no avail. I used $x$ to represent the number of seconds traveled with acceleration $f$ and then $t – x$ to represent the number of seconds decelerating. I'm still not sure how to do it. Any help would be appreciated.
Thanks.
Best Answer
The area under the curve equals the distance traveled because integratingis really just adding up every value of the curve. The slope of the left side it $f$ and the slope of the right side is $-r$. $$s=\mbox{area}=\frac12v_mt$$ Now lets find $v_m$ in terms of $f$ and $r$
Note that the slope is rise over run. $f=\dfrac{v_m}{x}$ and $r=\dfrac{v_m}{t-x}$
So we need to eliminate $x$ and find $v_m$$$x=\frac{v_m}{f},x=t-\frac{v_m}{r}$$$$\frac{v_m}{f}=t-\frac{vM}{r}$$$$rv_m=frt-fv_m$$$$v_m=\frac{fr}{f+r}t$$$$s=\frac12\left[\frac{fr}{f+r}t\right]t$$$$s=\left[\frac{fr}{f+r}\right]\frac{t^2}{2}$$$$s=\frac{\frac{frt^2}{f+r}}{2}$$