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Engine of a train that is moving with un...

Engine of a train that is moving with unifrom acceleration passes a pole with speed `u` while the last compartment passes the pole with speed 'v'. The middle point of the train passes the given pole with speed:

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Let x be the total length of the train, v be the velocity of the train while passing a certain middle point and a be the uniform acceleration of the train. Taking the motion of the train when middle point is passing from the given point, we have
`u=u, v=v, s=(x)/(2), a=a`
using `v^(2)=u^(2)+2as,` we have `v^(2)=u^(2)+2a(x)/(2)=u^(2)+ax" ...(i)"`
Taking the motion of train when the last end of train is passing from the given point, then
`u=u, v=v, a=a, s=x`
Now, we have `v^(2)=u^(2)+2ax or ax=(v^(2)-u^(2))/(2)`
Putting this value in (i), we get `v^(2)=u^(2)+(v^(2)-u^(2))/(2)=(u^(2)+v^(2))/(2)`
`rArr" "v=sqrt((u^(2)+v^(2))/(2))`
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