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A point traversed half the distance with...

A point traversed half the distance with a velocity `v_0`. The remaining part of the distance was covered with velocity `v_1` 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.

A

`(V_(0) (V_(1) + V_(2)))/(3(V_(1) + V_(2) + V_(3)))`

B

`(3V_(0) (V_(1) + V_(2)))/(V_(1) + V_(2) + V_(3))`

C

`(V_(0) (V_(1) + V_(2)))/(V_(1) + V_(2) + 4 V_(3))`

D

`(3V_(0) (V_(1) + V_(2)))/(V_(1) + V_(2) + 4 V_(0))`

Text Solution

Verified by Experts

Let s be the total distance . Let (s/3) distance be
covered in time `t_(1)` while the remaining distance (2s/3) in time `t_(2)` second.
Now `((s)/(3)) = v_(0) t_(1) " or" t_(1) = (s) /(3v_(0))` ……(A)
and ` (2s/3) = v_(1) ((t_(2))/(2)) + v_(2) ((t_(2))/(2))`
or `t_(2) = (4s)/(3 (V_(1) + V_(2))` ..... (B)
Average velocity `= (s)/(t_(1) + t_(2)) = (s)/((s)/(3V_(0))+(4s)/((V_(1) + V_(2)))) = (3V_(0) (V_(1) + V_(2)))/(V_(1) + V_(2) + 4V_(0))]`
Hence correct answer is (D)
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