Calculating power as a function of time

1. Nov 19, 2013

hk4491

1. The problem statement, all variables and given/known data
A force of 5 N acts upon a body of 8 kg in the +x direction. Formulate an expression for the power generated as a function of time. The body is in the beginning at t=0 and x=0 (Sorry I'm translating this question from German, so excuse the mistakes).

2. Relevant equations

Work = Force * distance

Power = ΔW/Δt

Force = mass * acceleration

3. The attempt at a solution

Since the force of 5N is acting on the body, by Newton's third law, it is also exerting the same force, so that:

F = m * a

5 = 8*a

a=5/8 m/s^2

By integrating the acceleration twice I get the distance as a function of time:

x(t) = 5/8 t^2

Work = Force * distance = 5 * (5/8)t^2 = (25/8)t^2

P(t) = (25/8)t

Was my method correct?

2. Nov 19, 2013

haruspex

You forgot something in doing that integration.
You have divided the total work done over the distance by the total time taken. That will give you average power, but I think it is the instantaneous power at time t that is wanted.

3. Nov 20, 2013

hk4491

Thanks I realised my mistake in the integration part. That leads to x(t)= 5/16 t^2.

The body starts moving at t0=0, so at any given point of time Δt = t - t0 = t

Hence:

P= (F*d)/t

P(t) = 25/16 t

4. Nov 20, 2013

Curious3141

No. You missed out a factor of half in your integration, so the correct expression for total work done at time t should be $W(t) = \frac{25}{16}t^2$. This expression is not linear in t, so can you simply divide by t to find the power at time t? Shouldn't you be differentiating?

Basically, your original expression was correct - simply because your errors "cancelled out". You missed a half when integrating, then divided instead of differentiating, so you didn't include a requisite factor of 2.

5. Nov 20, 2013

hk4491

Great, thanks for the explanation!

6. Nov 20, 2013

Staff: Mentor

Another way of arriving at the answer is to remember that the instantaneous power is equal to the force times the velocity. The velocity at time t is 5t/8.