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A body of mass $m$ moving along a straight line covers half the distance with a speed of $2 \mathrm{~ms}^{-1}$. The remaining half of the distance is covered in two equal time intervals with a speed of $3 \mathrm{~ms}^{-1}$ and $5 \mathrm{~ms}^{-1}$ respectively. The average speed of the particle for the entire journey is
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The correct answer is:
$\frac{8}{3} \mathrm{~ms}^{-1}$
Let the total distance travelled by the body is $2 S$. If $\mathrm{t}_{1}$ is the time taken by the body to travel first half of the distance, then
$\mathrm{t}_{1}=\frac{\mathrm{S}}{2}$
Let $\mathrm{t}_{2}$ be the time taken by the body for each time interval for the remaining half journey.
$\therefore \quad \mathrm{S}=3 \mathrm{t}_{2}+5 \mathrm{t}_{2}=8 \mathrm{t}_{2}$
So, average speed $=\frac{\text { Total distance travelled }}{\text { Total time taken }}$
$=\frac{2 S}{t_{1}+2 t_{2}}$
$=\frac{2 S}{\frac{S}{2}+\frac{S}{4}}$
$=\frac{8}{3} \mathrm{~ms}^{-1}$
$\mathrm{t}_{1}=\frac{\mathrm{S}}{2}$
Let $\mathrm{t}_{2}$ be the time taken by the body for each time interval for the remaining half journey.
$\therefore \quad \mathrm{S}=3 \mathrm{t}_{2}+5 \mathrm{t}_{2}=8 \mathrm{t}_{2}$
So, average speed $=\frac{\text { Total distance travelled }}{\text { Total time taken }}$
$=\frac{2 S}{t_{1}+2 t_{2}}$
$=\frac{2 S}{\frac{S}{2}+\frac{S}{4}}$
$=\frac{8}{3} \mathrm{~ms}^{-1}$
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