Consider a particle moving along x-axis. Its distance from origin O is described by the co-ordinate x which varies with time. At a time `t_(1)`, the particle is at point P, where its co-ordinate is `x_(1)` and at time `t_(2)`, the particle is at point Q, where its co-ordinate is `x_(2)`. The displacement during the time interval from `t_(1)` to `t_(2)` is the vector from P to Q, the x-component of this vector is `(x_(2) - x_(1))` and all other components are zero.
It is convenient to represent the quantity `x_(2) - x_(1)` the change in x by means of a notation `Delta`, thus `Delta x = x_(2) - x_(1)` and `Delta t = t_(2) - t_(1)`.

The average velocity `bar(V) = (x_(2) - x_(1))/(t_(2) - t_(1)) = (Delta x)/(Delta t)`
The resistive force suffered by a motor boat is proportional to `v^(2)`, where `v` is instantaneous velocity. The engine was shut down when the velocity of the boat was `v_(0)`. Find the average velocity at any time `t`.

Options

A. `(v_(0) + v)/(2)`
B. `(v v_(0))/(2 (v_(0) + v))`
C. `(v v_(0) log_(e) ((v_(0))/(v)))/((v_(0) - v))`
D. `(2 v v_(0) log_(e) ((v_(0))/(v)))/((v_(0) + v))`

Answer

C

Explanation

`F prop v^(2)`
`F = -Kv^(2)`
`m (dv)/(dt)` or `mv (dv)/(ds) = - kv^(2)`
`rArr v (dv)/(dt) = - cv^(2)`
`rArr int_(v_(0))^(v) (dv)/(v) = - C int_(0)^(s) ds`