Point traveling at different speeds

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Problem:

A point traversed half the distance with a velocity $v_0$. The remaining part of the distance was covered with velocity $v_l$ for half the time, and with velocity $v_2$ for the other half of the time. Find the mean velocity of the point averaged over the whole time of motion.


 Solution:

Let $d$ be the total distance traveled by the point and let $t_1$ be the time taken to cover half of the distance. Then, let $2t$ be the time taken to cover the rest half of the distance. 

Then, we have $$\frac{d}{2}=v_0t_1 \implies t_1=\frac{d}{2v_0}$$ and $$\frac{d}{2}=(v_1+v_2)t \implies 2t=\frac{d}{v_1+v_2}.$$

So, the mean velocity of the point is $$\frac{d}{t_1+2t}=\frac{d}{\frac{d}{2v_0}+\frac{d}{v_1+v_2}}=\boxed{\frac{2v_0(v_1+v_2)}{v_1+v_2+2v_0}}.$$

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