# Does the effect change if a continuing and constant force acts upon a mass?

If a continuing and constant force $F$ acts upon a body with the mass $m$, does the power (work per unit time) increase over time? It would mean that the converted energy per time unit increases. Is this true?

$W = Fs = F \cdot (\frac{at^2}{2}) = F \cdot (0.5 \frac{F}{m} t^2) = 0.5 \frac{F^2 t^2}{m}$

This gives $P_{\text{average}} = \frac{W}{t} = \frac{0.5 \frac{F^2 t^2}{m}}{t} = 0.5 \frac{F^2 t}{m}$.

Also, its derivative $\frac{d}{dx} 0.5 \frac{F^2 t^2}{m} = P_{\text{moment}} = 0.5 \frac{F^2 t}{m}$.

For me, this seems just absurd. This would mean that we can't really talk about energy per unit mass of a fuel, because the energy converted per second would depend on the current speed. Is there anything obvious I'm missing here?

## 2 Answers

You are assuming that your power source can supply a constant force $F$ while outputting a constant power. But you have already shown in the question this is not right, because the power outputted by the source is $F v$ where $v$ is the speed of the thing being pushed.

• Let's say that the force is gravity. Do the power of gravity increase with the velocity? – Madde Anerson Dec 11 '14 at 5:55
• Yes, becuase $P=Fv$ and $v$ is increasing. Also you can see the power from the change in potential energy. The potential energy is $mgh$, so if we view this as an energy source, it is being used up at a rate $mgv$, which is increasing as $v$ increases. So the constant $F$ is not being generated by constant power, but by an increasing power, as stated in my answer. – Brian Moths Dec 11 '14 at 13:16

The mass will not move at constant speed, it will be accelerated because of the Force. As the mass accelerates, it gets more work from the same force, that is, the power increases (remember: $P=F.V$). This power is giving an accelerated amount of work. Even if the force is constant, the power it produces becomes unbounded until it the speed becomes relativistic.

• But how can we talk about how much energy some fuel has per mass unit if the produced work if dependent on the velocity? – Madde Anerson Dec 11 '14 at 6:29