Average Speed

Question

A particle covers half of its total distance with speed ${v}_{1}$ and the rest half distance with speed ${v}_{2}$. Its average speed during the complete journey is

Moderate

Solution

Let *S* be the total distance travelled by the particle. Let ${t}_{1}$ be the time taken by the particle to cover first half of the distance. Then ${t}_{1}=\frac{S/2}{{v}_{1}}=\frac{S}{2{v}_{1}}$ Let ${t}_{2}$ be the time taken by the particle to cover remaining half of the distance. Then ${t}_{2}=\frac{S/2}{{v}_{2}}=\frac{S}{2{v}_{2}}$

Average speed,

${v}_{\mathrm{av}}=\frac{\text{Total distance travelled}}{\text{Total time taken}}$

$=\frac{S}{{t}_{1}+{t}_{2}}=\frac{S}{\frac{S}{2{v}_{1}}+\frac{S}{2{v}_{2}}}=\frac{2{v}_{1}{v}_{2}}{{v}_{1}+{v}_{2}}$

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