Question

A runner hopes to complete the 10,000-m run in less than 30.0 min. After exactly 25.0min , there are still 1900m to go. The runner must then accelerate at 0.25m/s2 for how many seconds in order to achieve the desired time?
Express your answer using two significant figures.

Answer 1

Runner passed 8100 m in 25 min = 1500 s. Hence, his speed was $v_{0}=8100/1500=5.4$ m/s Now, he have to pass S=1900m in T=5 min = 300s. Let us suppose, he will be accelerating t seconds. Then, equation will look like:

$v_{0}\cdot t_{1}+a\cdot t_{1}^{2}/2+(v_{0}+a\cdot t_{1})\cdot t_{2}=S$

with condition $t_{1}+t_{2}=T$, where $t_{1}$ is time when he moves with acceleration and $t_{2}$ is time when he moves at reached speed. We can solve this system for $t_{1}$:

$t_{2}=T-t_{1}$

$v_{0}\cdot t_{1}+a\cdot t_{1}^{2}/2+(v_{0}+a\cdot t_{1})\cdot(T-t_{1})=S$

$v_{0}\cdot t_{1}+a\cdot t_{1}^{2}/2+T\cdot v_{0}+a\cdot t_{1}\cdot T-v_{0}\cdot t_{1}-a\cdot t_{1}^{2}=S$

$-t_{1}^{2}\cdot a/2+a\cdot t_{1}\cdot T+T\cdot v_{0}-S=0$

$t_{1}=3.76\,s$

Runner has to accelerate for 3.76 seconds.

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