Say it takes 10 feet for an object to reach equilibrium speed when a constant force is applied to it. If that same constant force is then applied in the opposite direction to stop the object, will it take 10 feet for the object to come to a rest? And are the acceleration and deceleration processes identical absolute values when compared across time?
“equilibrium speed”?
I guess what you call equilibrium speed is the speed at which friction precisely equals the force being applied to accelerate the object? Then if the original force is discontinued and an equal force is applied in the opposite direction, the object will need less than 10 additional feet to stop: friction is also slowing it down, isn’t it?
Just my two cents…
equilibrium speed - when the object reaches a constant speed after getting past its inertia period.
“Inertia period”?
I don’t actually see any way to answer the question: If we’re assuming a simple dry kinetic friction, then there is no equilibrium speed, since if the applied force is ever enough to overcome the friction, it’ll always be enough to overcome it, and the object will accelerate indefinitely. If we’re assuming some sort of fluid friction that scales with speed, on the other hand, there will be an equilibrium speed, but the object will only asymptotically approach it, so the distance traveled before reaching equilibrium speed will be infinite.
Why do you think it will reach a constant speed?
An applied force gives an acceleration, which will continue to speed the object up for as long as the force is applied. Unless you mean that the “force is applied” only for an “instant”, and not continuously while the object is moving.
To partially answer your question, though : If there are no other similar forces, it will take just as much time and space to slow down as to speed up to any given speed, (within reasonable limits of the universe, anyway).
Friction counts as “another force”, but I’m not sure if you intended friction to be taken into account. By “similar force” I mean in the same or an opposing direction.
And with this “clarification”, the question becomes even more muddled. Inertia always applies to everything. “After getting past its inertia period” would mean “wait until the laws of physics change”.
I don’t understand the reasoning that leads to indefinite acceleration. If I push a cardboard box across the floor with some arbitrary constant force (dry kinetic friction) it has a constant velocity and constant frictional force.
As to the OP, if a force is applied to an object to move it against dry kinetic friction and then you apply the same force in the opposite direction, I would think it would stop instantaneously. If you simply stop the initial force, the object will stop in 10 feet just from the friction.
None of which has anything to do with inertia. Inertia is the resistance of mass to being moved. A force applied to a mass in the absence of friction will cause constant acceleration, with no equilibrium speed.
No, it’ll accelerate, at least assuming your pushing with more force then friction exerts in the other direction. Thats pretty much the definition of a “constant force”, it causes a constant acceleration, and thus not a non-constant velocity.
I assure you that it doesn’t have a constant velocity. Net force is equal to mass times acceleration, and so for an arbitrary applied force there will be some kind of acceleration. If you apply exactly the amount of force required to cancel out the force of friction, then the net force is zero, the acceleration is zero, and you have a constant velocity. But if you’re initially pushing the box with this particular amount of force, and then you decide to increase the force, the velocity will increase with time until such time as you decrease the force again.
ETA: Or, what Simplicio said.
I’ve got a feeling that the OP wants us to ignore friction altogether, but is perhaps under the impression that a constant force will only accelerate an object to a given velocity, after which the force would cause no further increase in velocity.
It doesn’t (for the most part), if you’re actually applying an *arbitrary *force. This is one of those situations where your physical intuition fails. When you push a box across the floor, you’re really applying whatever force is necessary to exactly counterbalance the friction and maintain your walking speed. If you run instead of walk, you’re still (for the most part) applying whatever force is necessary to exactly counterbalance the friction and maintain your running speed, and it’s essentially the same force you apply at a slower speed. It might feel like you’re applying a much greater force, but you’re not. You’re doing more work, in both the formal and informal sense of the word, because you’re velocity is greater, but it’s not (for the most part) more force.
I’ve persistently added parenthetical qualifications (for the most part) because, while dry friction is modelled with reasonable accuracy as being independent of velocity, that’s not 100% true. However, for discussion purposes, it’s close enough to true.
No it won’t. The box still has inertia, and it will take some distance to stop. As a thought experiment, imagine reducing the friction force on your box by putting it on wheels. In that case, it clearly doesn’t stop instantaneously, even when applying a force in the opther direection.
Accelerations and decelerations occur due to the total force applied to an object. There’s no reason why the friction force alone (the force that decelerates your box). would be exactly the same as the applied initial force minus friction (the force that accelerates your box).
Imagine that you put a spring scale between you and the box. The scale will register more 'weight" when you are accelerating the box. Once you reach the speed you desire and stop accelerating the box (i.e., a constant speed) the weight registered by the spring scale would be less. This means you are not applying a constant force throughout.
Ok I was basically thinking about a glass on a table, and got myself into some muddy water trying to generalize everything.
Bring your left hand towards the glass with some constant force…the glass will initially move slowly and then it will eventually reach a constant speed (when the friction and the force balance out). To clarify, your left hand is in constant contact with the glass.
Now stop the glass with your right hand with a constant force EQUAL to the previous force of the left hand (which is still in contact with the glass)…the glass will initially slow down slowly and then it will eventually reach a stop. To clarify, the right hand stays still and the glass comes to IT as it is pushed by the left hand, and the right hand has a steady force of resistance.
Does this process isolate the speed-up inertia and the slow-down inertia? I’m trying to visualize the parallel between starting and stopping something on a flat surface; there seems to be an isomorphism there.
This isn’t true. If you push something with a constant force, friction or no friction, it will never reach a constant speed, it will accelerate until you reduce the force on it. In real life when you want to push something at a constant speed, you decrease the force once you have it moving at the speed you want it to go so that your just barely pushing it enough to overcome friction.
First of all, what Chronos said:
That’s not quite true, since even dry friction has some velocity dependence, but it’s close enough to true that it muddles your thought experiment.
However, let’s ignore that, and simply assume that you’re pushing the glass with some force (which is perhaps variable) that accelerates the glass up to a certain speed and then holds the glass at that constant speed. Since the speed is constant, then by definition there’s no acceleration, and if there’s no acceleration there’s no *net *force (since F = ma = 0). No net force means the force from your hand is exactly canceled by the friction force.
The “inertia” is the mass of the glass; there’s no differentiation between “speed-up inertia” and “slow-down inertia.”
In any case, what happens is that the acceleration or deceleration will be a function of the net force on the glass. Just like when the net force is zero the glass maintains a constant velocity (F = ma = 0), when the net force is non-zero, the glass either accelerates or decelerates.
The key here is that the friction force is always in the opposite direction of the velocity. So when you’re trying to speed the glass up, the net force is (F[sub]hand[/sub] - F[sub]friction[/sub]), and when you’re trying to slow the glass down, the net force is (-F[sub]hand[/sub] - F[sub]friction[/sub])…meaning that the friction force is always acting to slow the glass down.
If the force of your hand is the same, then glass will slow down more quickly than it speeds up, because in one case the friction force reduces the net force, and in the other it adds to the net force.
That’s why I said simple dry friction. Real-life friction isn’t simple. But there’s a big enough jump in complexity from the nice simple constant-coefficient-of-friction model to the next more realistic model, that it’s almost never worth it to go to that extra complication.
Say I wanted to start the glass moving, get it up to a steady speed, and then stop it in the exact same way as I started it so that the entire process is exactly reflective across the time axis visually (imagine the hands are invisible and you only see the glass). How would the distribution of forces play out? Is it extremely complicated?
So, you want it to accelerate evenly up to a set speed, then decelerate at that same exact rate until it stops.
It depends on the amount of friction. I’ll give you 3 cases
#1) You apply 15 units of force to the glass, the glass generates 5 units of frictional force, so you accelerate the glass with 10 net units of force. When you decelerate the glass, in order to have the same rate of deceleration, you need to apply 10 net units of force to slow it down. That means applying 5 units of force in the opposite direction, so you have 10 net units to slow it down.
#2) You apply 10 units of force, the glass generates 5, you get a net 5 units of force accelerating the glass. To decelerate, you want 5 net units of force in the opposite direction, which you get by letting go of the glass, and letting friction do all the work for you.
#3) You apply 7 units of force, the glass generates 5, getting a net 2 units of force accelerating the glass. To decelerate, you want 2 units of force going the opposite direction, so you should reduce your force to 3 units, but keeping the same direction, to get the net of 2.