What is the Limiting Velocity of an Electric Toy Car with Constant Power?

[alice137]

Homework Statement

This problem concerns the mathematical treatment of a simple model
of an electric toy car of mass m, which is initially stationary. The bat-
teries in the car can be considered as an electrical power source with
constant power P.

Find the limiting value of the velocity at very large times and
comment on whether your result seems reasonable.

Homework Equations

On calculating I got v=sqrt(2Pt/m).

The Attempt at a Solution

As t -> Infinity, so does v-> Infinity. But that does not makes sense. Where am I wrong. Please help.

 

[alice137]

Another part of the question was:

Find the limiting value of the acceleration at very large and very
small times and comment on whether your results seem reason-
able.

On calaulating I got a=sqrt(p/2mt).

As t -> 0, a -> Infinity. Again That doesn't makes sense.

 

[tiny-tim]

Find the limiting value of the velocity at very large times and
comment on whether your result seems reasonable.
…
But that does not makes sense. Where am I wrong

Hi alice!

What makes you think you're wrong?

The question obviously expects the answer not to be make sense.

You're asked to comment on why correct maths doesn't seem to agree with reality!

What do you think … ?

 

[alice137]

Well I personally believe that the answer for acceleration makes sense. Initially the car might have acquired some velocity in a small time. I think the high acceleration is plausible.

For velocity, it cannot increase without bounds as velocity of a material object is limited by Relativity. But if we see the time dependent relation of acceleration and see that as t-> infinity, a->0, which means that v-> some constant. But I cannot find that constant based on these arguments.

 

[tiny-tim]

For velocity, it cannot increase without bounds as velocity of a material object is limited by Relativity.

Hi alice!

At relativistic speeds, the mass m gets bigger, and the original formula is probably wrong also.

But if we see the time dependent relation of acceleration and see that as t-> infinity, a->0, which means that v-> some constant. But I cannot find that constant based on these arguments.

No, you're wrong - there's no problem there. a->0 does not have the same consequences as a = 0: there's no reason why v should -> constant.

√n -> ∞, but √(n+1) - √n -> 0 (for example, √1,000,001 - √1,000,000 ~ .0005).

 

[alice137]

What do you insist then; v should increase without bounds?

 

[tiny-tim]

No, but the momentum can increase without bounds, even though v is always less than c.

(momentum = mv/√(1 - v^2/c^2).)

 

[alice137]

Then I suppose that the answer to the first question should be v-> c at very large times even although we cannot see it directly from the equation. I guess we need to derive another relativistic expression for very large times. Right!