# Why doesn’t a cart accelerate when I push it with a constant force?

If I push a cart with a constant force, it moves at a constant velocity, so there must be no net force. The applied force must be canceled out by kinetic friction and air resistance, but the magnitude of kinetic friction depends on the normal force, which doesn't increase together with the cart's velocity. So if the cart is moving at a constant velocity and I push harder, the constant force I apply should now be greater than the force of friction, as neither the normal force nor the friction coefficient changed. But we know from experience that pushing harder only results in a short period of acceleration, followed by constant velocity again.

So what am I missing? Is there a gap in my knowledge somewhere? Is the answer air friction, because its magnitude is proportional to the particle's velocity? (The same phenomenon is of course observed when you push any object. Say pushing a pen on the table, or letting a contraption push the pen with a constant force if we decide that forces applied by humans can never be truly constant. I don’t think the question is about the inconsistency of human biology, because a pushed object will always share its velocity with the pusher, no matter if it’s a person or a machine. This final nugget is especially confusing to me. Thank you!

• "But we know from experience that pushing harder only results in a short period of acceleration" what experience are you referring to? The cart should accelerate. Commented Apr 28 at 14:38
• Friction usually depends at least linearly on the speed at which the object drags through a medium. The friction force applied is most certainly NOT a constant. Commented Apr 28 at 15:27
• Measuring force with your own body isn't a very reliable method, especially if your body is constantly moving becasue your brain have to deal with keeping your body in motion. You may also apply force in the wrong direction i.e. you pushed the cart downwards instead of horizontally. Tire also gives false perception Commented Apr 28 at 15:41
• @BobD is incorrect here. There is a very valid reason for an object to stop accelerating without you reducing your force, unconsciously or not, and that is simply that net force returns to zero by any number of counterforces: friction, drag, restoring forces, etc. Commented Apr 30 at 19:23
• @jazzblaster This is similar to an object achieving a terminal velocity in free fall. Gravity exerts a constant acceleration and drag counters to achieve constant velocity. Commented Apr 30 at 22:48

You actually answer your own question quite nicely, you aren't missing anything. The answer is air resistance (or an equivalent form of drag). You are also right to dismiss the comments attributing the effect to some human or biological factor, indeed a contraption that can reliably apply constant force would observe the same thing, an engine or rocket for example.

Any counterforce on an object that is proportional to that object's velocity will cause the effect you describe. You push an object with a constant force, causing it to accelerate for some time. It increases its velocity, increasing the counterforce, until it reaches a velocity where that counterforce is exactly equal to your applied force, at which point it will stop accelerating and maintain its velocity. Think a car on a highway. At a given engine throttle, it is applying a constant forward force, yet maintains highway velocity because at that speed, air resistance is proportional to velocity, squared, and the speed your car settles in that speed which produces a drag force which exactly counters the forward force from the engine.

And it is not just air resistance that applies a counterforce proportional to velocity. In fact, static and dynamic contact friction is sort of an outlier in this regard. Damping from internal forces in a material, electromagnetic energy loss, viscous fluid drag, etc. all also have this effect. So the effect you describe will almost always occur in the real world, despite the fact that in ideal land we are taught a constant force leads to constant acceleration.

• "The answer is air resistance (or an equivalent form of drag)" I have a hard time believing that air resistance plays any significant role here, considering the likely speeds involved. The "biological factors" are probably the most important. Commented Apr 30 at 19:43
• @BobD As I've responded before, the biological factors are not important here whatsoever, and the question poster has also asked to disregard them as well. You will notice the same effect with an engine that can output a constant force, or even the Earth which close to the ground exerts a constant force, eventually a constant velocity will be reached. This is the entire concept behind terminal velocity, for example. Commented Apr 30 at 22:45
• @BobD You are at least right in saying that in the very specific cart case, it is not air resistance, but it is an equivalent form of drag. There are velocity-dependent sources of energy dissipation when pushing a cart which eventually match the constant pushing force, causing acceleration to cease. For example non-linear losses in the wheels of the cart. That is what I meant by equivalent, something with a similar dependance on velocity. Commented Apr 30 at 22:50

This is actually a very good question and when we look at the standard equations for friction and ignore air friction which is negligible at walking speeds, the situation is actually paradoxical as you suggest. Static friction, kinetic (sliding) friction and rolling friction are all ostensibly independent of velocity. This implies that if a constant force is applied that is slightly greater than the rolling friction of the trolley, then it should keep accelerating at least until air friction does become significant around 50 mph. How is this resolved?

The text book formula for rolling resistance ($$r_R$$) on a flat horizontal surface is $$r_R = \mu mg$$ which is a constant, but this is an idealised approximation and in reality, it is likely that the the formula should have an additional small term that is a function of velocity and most likely a function of velocity squared. Support for this assumption comes from some experimental tests of the relationship between friction and angular velocity of a disk.

In our everyday experience, e.g. when pushing a shopping trolley around a supermarket, we might think we are applying a constant force, but as you have already worked out, that would result in a constant acceleration and the trolley would reach a speed that exceeds walking pace and then we have to start running faster and faster to maintain that constant force until we reach a speed where we physically cannot run any faster, so clearly we are not maintaining a constant accelerating force and subconsciously we back off to a comfortable pace and a force that matches the rolling resistance.

This is somewhat akin to the situation where a human holds a weight with their arm outstretched parallel to the floor. Since the weight is not being raised the total work done and energy required to hold the weight at constant height is theoretically zero. In practice we know it does take effort to hold the weight at constant height. This is because our muscles contract and relax periodically in turns and this produces a small almost imperceptible oscillation and the weight is not really at being held at constant height but constantly falling and being raised back up around an equilibrium height and so energy is expended.

In summary, it seems that the standard equations in school text books are an approximate idealisation of reality and do not tell the full story and neither does our subjective everyday experience. A closer description of reality will be found in engineering text books and research papers.

Firstly, your example of a cart may not be of constant force. I assume your cart example is the shopping cart we use and walk around with. 'Walking around' with a cart and 'applying constant force' on a cart are two different things. When you are walking around with your cart, you are holding it. This 'hold' results in a force directed towards you, which cancels out the force you apply for its translational motion. Even when you push a pen on a table and the pen moves at a constant velocity with your finger, you are, unknowingly, applying a constant force opposite to the motion of the pen, on your finger. You can experience this force if you concentrate on your hand. You would feel that it is 'holding your finger' back. It's just our human tendency. When a human (or any other life form) pushes, it does not necessarily mean they are applying constant net force. It's also true that when constant force, which you have applied, equals frictional force, the body will attain a constant velocity. But, again, only when you are applying a constant force that has a specific magnitude.

Say pushing a pen on the table, or letting a contraption push the pen with a constant force if we decide that forces applied by humans can never be truly constant.

So let's talk about actual 'constant force'. According to Newton's second law of motion for a body with 'constant mass', constant net force ($$\neq0$$) results in constant 'acceleration' ( $$\vec{F} = m\vec{a}$$ ). Constant acceleration means velocity is 'changing' constantly. So for any type of constant force (which is not 'balanced' by any other opposite force), velocity has to change constantly. The best example, in my opinion, would be a freely falling body. For not-so-significant heights, the gravitational force is constant and thus freely falling body experiences constant acceleration. But practically, for sufficient height, a body attains a constant terminal velocity, due to air resistance or drag force.

If I have missed anything, I apologise in advance. Hope this helps.

It is very difficult to supply anything close to a constant force with your hands. If you really want to test this, I'd suggest using a rope with a force scale on it for tension, and use that to pull the cart. So my first explanation of your experience is that the pushes are not with constant forces.

the magnitude of kinetic friction depends on the normal force, which doesn't increase together with the cart's velocity.

This is mostly true, but is an idealization. There are lots of potential sources of drag that are not uniform with speed, even without getting into air drag. Cart wheels can get vibrations at certain speeds that create significant drag that is not present at slower speeds.