final velocity to initial velocity ratio

Question

A particle of mass m, moving with velocity vi has kinetic energy equal to Ki. If work W = 8Ki is done on the particle, the ratio of the final velocity of the particle to the initial velocity of the particle, vf/vi is:

Answer 1

The answer is that "It is impossible to tell".

The reason is that the energy gained from the work might be totally or partially

in in the form potential energy.

That makes the change in kinetic energy and speed unpredictable.

I will answer this question modified in this way:

"If work W = 8Ki is done on the particle without changing its potential energy ...".

With this modification all the work W is converted to an increase in kinetic energy.

By definition of kinetic energy

Ki = mvi²/2

Kf = mvf²/2

By our assumption that all the work W gets converted to kinetic energy

Kf = Ki + 8Ki = 9Ki

Divide the equation

9Ki = mvf²/2

by the equation

Ki = mvi²/2

to get

9 = vf²/vi²

Therefore

vf/vi = 3