Why Is Speed Proportional to the Square Root of Time in This Physics Problem?

[NewtonGalileo]

Homework Statement

A particle starts from rest and is acted on by a net force that does work at a rate that is proportional to the time t. The speed of the particle is proportional to:

sq root of (t)

t

t^2

1/sq root(t)

1/t

Homework Equations

power = work / time
work = change in kinetic energy

The Attempt at a Solution

So,
(1/2 mv^2 / t) proportional to (t)
So,
v proportioanl to t

But, the answer key says v proportional to sq root of t. How is that?

 

[collinsmark]

power = work / time
work = change in kinetic energy

The Attempt at a Solution

So,
(1/2 mv^2 / t) proportional to (t)

Be careful here. Power is also equal to force times velocity.

In this particular problem, the force is proportional to t, as you know.

But the power is also proportional to velocity. But what is velocity?

Make the substitutions into force and velocity, and the answer should become more clear.

 

[NewtonGalileo]

But,

net force is a constant. It is not proportional to time t.
I also looked at:
power = work/time = force x velocity
power ~ t
force x velocity ~ t
since force is a constant,
velocity ~ t

wouldn't this be right? why sq.root of t?

 

[collinsmark]

Perhaps you're right. I think I originally misinterpreted the problem statement as saying the force was proportional to t. But now I see it says that the rate of work is proportional to t. So yeah, that makes a difference.