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For a constant force, P=Fv. I understand the mathematical derivation of this, but this seems to me, intuitively, to be nonsense. I feel that my discomfort with this comes from a fundamental misunderstanding of force and Newton's Second Law, so I'm not really looking for any mathematical explanation. So, to begin:

How is it that a constant force does not add energy to a system at a fixed rate? Consider a rocket burning a fuel at a constant rate. The chemical potential energy should be converted to kinetic energy at a constant rate, that is, 1/2mv^2 should be increase linearly. The magnitude of the velocity of the rocket would then increase at a less than linear rate, implying a nonconstant acceleration and therefore, a nonconstant force/thrust (F=ma).

If force is indeed a "push or a pull," shouldn't that constant rate of burning of fuel yield a constant "push or pull" as well? Clearly not, so I would have to think that, somehow, a given force applied to a certain object at rest would in some way be different than that a force of the same magnitude being applied to that same object in motion. In this sense, is force merely a mathematical construct? What does it tangibly mean, in physical terms? Would a given force acting upon me "feel" differently to me (in terms of tug) as I am moving at differing velocities?

Force being defined as a "push or pull," which is how it has been taught in my high school class, seems rather "handwavy," and maybe that's the issue. It's been troubling me for a couple of weeks and my teacher hasn't really been able to help, so thanks!

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How is it that a constant force does not add energy to a system at a fixed rate?

Because the velocity isn't constant. Think of it this way; the force is constant but the distance through which the force acts, per unit time, and thus the amount of work done by the force, is changing.

For the energy to change at a fixed rate (for the power to be constant), the work done per unit time must be constant; the force would need to decrease in inverse proportion to the speed.

As an aside, in the case of a rocket, you must also consider the energy of the exhaust products, i.e., the PE of the propellants is converted to KE of both the rocket and the expelled combustion products.

Also, since the rocket is expelling mass, the acceleration of the rocket, for a constant thrust, will not be constant

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I understand the "quasi-mathematical" explanation that since velocity isn't constant, energy is not added at a fixed rate. What is hard for me to swallow is that the instantaneous velocity should have bearing on the amount of work the force/"push or pull" is doing at all. I mathematically get that W=Fd, but it seems to me that a constant force/"push or pull" should still change the energy of a system at a constant rate. Why should a "push or pull" be defined in terms of acceleration and not energy change; once again, what really is a force in physical, not mathematical, terms? –  high schooler Nov 18 '12 at 1:49
    
Consider a linear electric motor pushing a mass with a constant force. In the 1st second, the motor moves through 1 meter. In the 2nd second, the motor moves through 3 meters and in the 3rd second, through 5 meters. Is it not intuitive that the motor does more work in the 2nd second than the 1st? And in the 3rd than the 2nd and so forth? –  Alfred Centauri Nov 18 '12 at 2:00
    
The mathematics of F=ma and W=Fd yield that conclusion pretty intuitively. Physically, it should make more sense that a constant tug (force) should change the energy of a system linearly rather than the velocity. Otherwise, the amount of "tug" I would feel on myself when a constant force(=ma) changes my motion would not be constant. –  high schooler Nov 18 '12 at 2:09
    
I don't think it should make more sense physically that the energy should change linearly; momentum yes, energy no. –  Alfred Centauri Nov 18 '12 at 2:13
    
I'm probably not being very clear. Essentially, that a constant force can output a nonconstant power makes sense when considering them to be mere mathematical constructs. But it makes no sense to me that they should be defined as such! Momentum, to me, seems to be even more of an arbitrary construct (which happens to be usefully mathematically); what is momentum in physical terms? The most natural definition of force, I feel, should be of a constant power output. –  high schooler Nov 18 '12 at 2:21
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I know what you mean. "Force" is quite a strange concept. Some thoughts:

F * x = E = W (if you think in one dimension)

Force applied along a way is energy.

If you want something that you can apply over time and get energy, you are looking for power.

I would have to think that, somehow, a given force applied to a certain object at rest would in some way be different than that a force of the same magnitude being applied to that same object in motion.

That is kinda correct. The faster the object is moving, the more power is applied: F * v = P

Getting back to your rocket example, where the engine burns fuel at a constant rate. So, the POWER of the rocket machine is constant, power * time = energy. Also, force * speed = power. That means as the speed increases, the force that the engine applies to the rocket slows.

As you said, the rocket should gain kinetic energy linearly with the time: E_kin = enginePower * time. Since E_kin = factor*v^2, we get v^2 is proportional to the time, which in turn gives v = somefactor * sqrt(time). The speed is propotional to the square root of the time. Since force times speed should be constant, the force is proportional to the inverse of the square root of time.

In other words: Accelerating at high speeds costs more energy than at low speeds. If you push on something that is fast (you apply force), you will waste more Energy doing so because you do more way, do give a Rocket constant acceleration, you must burn more and more fuel.

"Pushing" or "pulling" for a human being is always connected to energy consumption, even when you push a resting object. (It is something with our muscles, I think.) Force isn't connected to energy consumption. When you think about pushing, you probably think rather about applying power than about applying force.

PS:

I know these kind of questions, when you think about something, and the more you think about it, the less it makes sense. And then you try to ask somebody who should know, but they don't understand your problem, and then you wonder if they are all stupid. There is also the other kind of questions where you know you are right and they are wrong, but nobody wants to hear it. It can be frustrating.

PPS:

There is some barrier between the physics and the things we experience. You take a situation translate it to a formula, do some math and translate it back. This translation process is blurry. You can write a lot about the interpretation of a theory (here, we have been interpreting classical mechanics), but you will use words for that, and words are not precise.

It is a very good thing that you try to get a non-abstract understanding of the basic physical laws. You might be on the way to become a good Physicist.

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So in the end, would you agree that as human observers, we don't naturally "feel" forces=ma, but rather, we "feel" power more easily instead? I think you understand what I am asking and what I am feeling. My teacher did not seem to understand when I asked him, so I appreciate it! –  high schooler Nov 18 '12 at 2:46
    
I am not sure, to be honest. –  Konstantin Nov 18 '12 at 17:49
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Consider objects in a constant gravitational field. That is, for any object of mass $m$, there is a constant force field $|F| = mg$ directed downwards, toward the earth. There is then an associated potential energy $U = mgy$ for a distance $y$ from the surface of the earth. Any object that moves vertically by 1 meter gains a fixed amount of energy regardless of where they started from. This is what characterizes a constant force field.

So (one thing) force tells us how potential energy changes with position. If one makes a straight vertical path from $y=0$ to $y=h$ for some height $h$, it should not yield a different potential energy than taking a very circuitous, meandering route. Each position has exactly one value for the potential energy, and that's all.

Now, consider two objects that travel from the height $y = h$ to $y=0$. The potential energy difference is $\Delta U = mgh$. Let object $A$ start from rest at $y=h$. Let $B$ have some downward velocity. Clearly, $A$ will lose energy less quickly than $B$, for it takes $A$ longer to reach the ground.

That's why velocity affects power gained or lost. Energy losses in a force field depend only on how that force changes with position. If positions are traversed more quickly, then any changes must occur more quickly.

This line of reasoning depends on the notion of fields, rather than forces from things other than fields acting on objects. Nevertheless, it is rare in physics that force explicitly depends on time (rather than depending on position, which in turn may or may not depend on time).

Finally, I urge you to think more closely about momentum, as it is a key concept in physics and more than just a handy quantity to use. Momentum is intricately tied to the concept of mass. If only velocities mattered, we would have no concept of inertial mass at all, for you could add objects velocities together blindly without regard to how much stuff there was. Mass serves to tell us that, more or less, heavier things matter more than lighter things. A heavy object moving slowly can matter just as much to a problem as a light object moving quickly. How momentum changes directly leads us to the notion of force.

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A constant force applied to an object at has the same 'effect' (in terms of acceleration) as it has on object moving at constant velocity, but different 'effect' in terms of kinetic energy. This is because the velocity of an object is relative to some (inertial) frame of reference. An object is deemed 'at rest' or 'moving with constant velocity' when measured with respect to some reference. An object 'at rest' has a constant velocity, namely, zero. This is just a statement of Newton's 1st law.

Kinetic energy is quite different from force. Kinetic energy depends on your frame of reference.

Suppose you're traveling in a spaceship with constant velocity though space. The kinetic energy of the spaceship is constant. Now you turn on the rocket boosters, hot gases are emitted at high velocity from the back of the rocket. Power is transmitted to the spaceship and you accelerate away further into space. As long as the rocket is switched on, you will experience acceleration. It will appear as if the power of the spaceship is constantly increasing!

But wait, if the rocket is applying a constant force, I can understand the kinetic energy of the spaceship increasing, but how is it that its power is increasing and not constant? Isn't the rocket a constant power machine!?

The reason for the apparent discrepancy is that if all of the rocket’s energy is transferred to kinetic energy of the spaceship, then the spaceship will accelerate. Also, as long as the force of the rocket is in in the same direction as the spaceship's instantaneous velocity, your speed will increase and so will your power!

To help your intuition get a grasp of this, think what would happen if you suddenly noticed your spaceship was heading for a large asteroid. If you don't switch off your rocket, the power of the impact will be huge! In fact, as long as your rockets keep thrusting your spaceship in the direction of the asteroid, the power of the impact is increasing. Even if you turn off your rocket, you will still be traveling at constant velocity toward the asteroid and will be doomed. The power of the impact is 'fixed' by your speed. In fact, to reduce the impact, you have to reduce your velocity (ie: accelerate in the opposite direction). Say you put two retro-rockets on full reverse-thrust, you will experience a force away from the asteroid, even though you are still traveling towards it, until at some point, you will appear stationary with respect to the asteroid before accelerating away from it.

Now, if you want to land on the asteroid, you must adjust the power of your retro rockets to reduce your speed such that at the speed is close to zero at the instant of the impact, otherwise your shock absorbers must be good enough to absorb the extra energy!

Constant force does not add energy at a fixed rate!

You will have noticed this when you try and push-start a car. When the car is stationary, your your force has little effect on the car. Once the car starts moving though, the force you apply is easily translated into kinetic energy! Of course, this is due to the momentum of the car. Its also easier to push an empty car from standstill than a full one. By the same token, more 'brake power' is needed to stop a heavy moving car (in a given time), the faster it is traveling.

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You made a really helpful point! The "velocity derivative" of kinetic energy is momentum, so it is greater momentum leads to greater power as we accelerate. Still, in the case of the rocket though, we are physically limited by the rate at which we can burn through our fuel. Thus, if power must then be constant, musn't force=P/v necessary drop off as we accelerate? –  high schooler Nov 18 '12 at 3:38
    
Basically, anyway, rockets/engines/propellers/whatever when pushed to their limits are limited by a constant, maximum power, correct? The force output is still clearly nonconstant, unless I am completely still missing the point. –  high schooler Nov 18 '12 at 3:42
    
That is correct. If the rocket/engine/propellers/whatever are providing constant power, then the force will decrease inversely with velocity (that is, component of force in the same direction as the object's velocity). –  theo Nov 18 '12 at 4:07
    
@highschooler, it occurs to me that the issue here may be that you are conflating the concepts of thrust and power. A rocket engine, for example, may develop a constant thrust but the power is split between the rocket and the exhaust products. For example, in the reference frame of the rocket, all of the power of combustion is delivered to the exhaust products. –  Alfred Centauri Nov 18 '12 at 4:20
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There's nothing wrong with any of these other answers, but for another perspective, if you have a constant force acting on an object starting with zero velocity, then it will accelerate with constant acceleration $\frac{F}{m}$, and thus, after $t$ time, will have velocity $v=\frac{F}{m}t$. This means that the kinetic energy that it has acquired will be given by $\frac{1}{2}mv^{2} = \frac{F^{2}t^{2}}{2m}$.

Since the power is the rate of energy consumption, we have:

$$P = {\dot E} = \frac{F^{2}t}{m}$$

so, it should be obvious that the power increases with time. It should also be clear that our expression for $P$ is equal to $Fv$.

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