$n$ equal distances

Calculus Level 3

If a particle travels $n$ equal distances with speeds $v_1,v_2,\ldots,v_n$ , find the average speed $v$ of a particle in terms of $v_1, v_2,\ldots, v_n$ .

v=\frac{v_1+v_2+\cdots+v_n}{n}

v=\sqrt{v_1^{2}+v_2^{2}+\cdots+v_n^{2}}

\frac{1}{v}=\frac{1}{n} \left(\frac{1}{v_1}+\frac{1}{v_2}+\cdots+\frac{1}{v_n}\right)

v=\frac{nv_1v_2\cdots v_3}{v_1+v_2+\cdots+v_n}

1 solution

Sparsh Sarode
May 15, 2016

We calculate the average velocity as the total distance over total time:

$v_{avg}=\frac{ns}{\frac{s}{v_1}+\frac{s}{v_2}+...+\frac{s}{v_n}}$

$\frac{1}{v_{avg}}=\frac{1}{n}(\frac{1}{v_1}+\frac{1}{v_2}+...+\frac{1}{v_n})$

n n n equal distances

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$n$ equal distances