# Motion of person in bus in inertial and non-inertial frame

A person is sitting in a bus that is moving with constant velocity, due to friction their is no relative velocity between bus and the person. Suddenly the bus starts to move with a constant accleration:

Inertial Frame of reference: For a person standing outside, they observe that it isn't the person being moved backward but rather the bus accelerates to a new velocity and the person due to Newton's 1st law continues to move with their original velocity until the normal force due to bus seat and the friction between them and seat accelerates them to new velocity of the bus. So if the bus has a constant acceleration $$a$$, the $$F_\mathrm{net} = ma$$ (on person in the bus)

Non-Inertial frame: For the same person sitting in the bus, initially for him, he is at rest. Then as bus starts to accelerate, the person is still acted upon by frictional force and the normal force but in his frame of reference he is not accelerating, so Newton's laws are invalid. Here is where the pseudo forces come into picture and act opposite to frictional and normal forces from bus so as to make net force zero thereby allowing us to apply 2nd law in the non-inertial frame of reference.

My doubt is, if the fictitious force is not a real force why do we still feel a force pushing us backward while sitting in accelerating car?, how can we feel a force if it isn't there? and is this the reason why fictitious forces are also called inertial forces as they do same thing as what inertia does in inertial frame? or is my understanding of fictitious forces wrong?

I am sorry if this question is too basic and sounds repetitive.

• See my answer to Coriolis Force: Direction Perpendicular to Rotation Axis Visualization. It is about a different fictitious force, but I explain why they are real. See also the link in it for your bus question. Commented Jul 11 at 14:50
• Unless I misunderstand you, something seems amiss regarding your scenario. If the person is seated, then the back of the seat would provide the necessary force to accelerate the person along with the bus. No friction would be required. Only if the person is standing and not holding on to something would friction be required. Commented Jul 11 at 15:00
• @mmesser314 so fictious forces are real in frame of reference of the person/bus because we consider the bus to be at rest and so for bus the person is being pushed backward with the same acceleration the bus is moving in inertial frame right? Commented Jul 11 at 15:00
• @BobD I am sorry but I actually didn't think much about it. Commented Jul 11 at 15:09

My doubt is, if the fictitious force is not a real force why do we still feel a force pushing us backward while sitting in accelerating car?

If you were somehow levitating in the bus as it accelerated, obviously you wouldn't feel anything. In your case when you accelerate with the bus, you feel the back of the seat / friction pushing you forward. You always "feel" a "push" opposite of your acceleration, similar to how you feel pulled outward around a turn. That's inertia for you!

You need to be very careful about what we mean by "fictitious". It doesn't mean "imaginary", "made up", etc. It just is a label we put on these "forces" because they do not follow Newton's third law. They don't arise from interactions; they come from the acceleration coordinate system in order for Newton's second law to still be valid in that accelerating coordinate system. So we call them "fictitious" in order to preserve "real" forces as ones that follow all of Newton's laws.

Alternatively, we can throw out the idea of "fictitious" forces altogether and modify Newton's second law to say that accelerations can arise without forces.

so fictitious forces arises because we consider ourselves as an inertial observer but it actually is just the inertia trying to maintain our velocity? is this why fictitious forces are called inertial forces?

Yes, that is why they are called "inertial forces". It really is a consequence of taking the inertial term $$ma$$ and treating it as a force in Newton's second law.

But to be clear, these fictitious / inertial forces aren't as tied to our "feelings"... If you are doing your analysis considering an accelerating reference frame and want Newton's second law to hold you will have to introduce such forces, even if the object isn't accelerating in an inertial reference frame. The "feeling" we get is just because accelerating makes us "feel" in the same way any other force would "feel" (note, this is one of the key concepts that leads to general relativity; acceleration being equivalent to being at rest in a gravitational field).

If you want to attribute what you "feel" to a "force", that is fine, it just isn't a force that adheres to Newton's third law.

• so fictitious forces arises because we consider ourselves as an inertial observer but it actually is just the inertia trying to maintain our velocity? is this why fictitious forces are called inertial forces? Commented Jul 11 at 15:31
• @HrishikeshPandey I replied to your comment in my edited answer Commented Jul 11 at 15:54

I think, If you are considering acceleration in any sense, you need to include the reference frame in which the acceleration is taking place. In 2nd person frame of reference, the one observing from outside, acceleration is observable because it is happening in this frame of reference(the car is accelerating with respect to environment). Whereas, In 1st person's frame of reference, the one in the car, considering acceleration would be invalid (because this is due to difference in momentum, person's momentum is less than that of car, so he feels that push opposite to the direction of motion), because you're not referring to the correct frame.

In Summary, to consider acceleration, right frame of reference should be used. Hope this helps.

What we feel in an accelerating vehicle is the force from the seat back that is pushing us forwards so that we accelerate along with the vehicle.

What we unconsciously assume is that there must be an equal and opposite force pushing us backwards into the seat. This is because we are used to being in a (more or less) inertial frame most of the time, so if we are apparently stationary we assume any force acting on us must be balanced by an equal and opposite force. For example, we are used to feeling the floor push upwards against our feet, which we assume (correctly) is an equal and opposite force to our weight.

• It definitely goes beyond assumptions though. If the bus is accelerating forward enough you will definitely find it harder to accelerate your hand forward relative to the bus, as if there is some extra force you have to fight against. Commented Jul 11 at 15:13
• @BioPhysicist Well, it's not a conscious assumption, so let's call it intuition if you like. Feedback from your muscles tells you that you have to push harder than usual to move your hand forward, so your intuition is this must be due to some force pushing backwards on your hand. But there is no mysterious force pushing backwards on your hand - it is just that the forces acting on your hand, like on your whole body, are not balanced - which feels unnatural. Commented Jul 11 at 15:21
• For sure! There is no extra "force that adheres to Newton's third law" pushing on you, at least Commented Jul 11 at 15:23
• oh ok so the reason for feeling of being pushed backward is because even though we are accelerating, in our frame of reference we are stationary so we assume their is an equal opposite force to that of the force the bus is applying on us? Commented Jul 11 at 15:27

These effects are often called ficticious forces. I'll often call them pseudo forces, but I have found the best way to think of them is as accelerations. When you do the equations of motion in a non-inertial frame, it is an acceleration that appear. This is a real acceleration, part of the equations of motion. It becomes "ficticious" or "pseudo" when we multiply it by mass, put it on the other side of the equation, and treat it like a force. Pseudoforces do not obey Newton's laws because there is no other object with which to have an equal and opposite reaction. (quite often I find that the search for an equal and opposite reaction to a pseudoforce leads people to link it to some other force in the system, like a normal force from the seat, and confuses the concept of equal and opposite pairs as defined in Newton's third law)

That being said, your body doesn't feel forces directly. It feels the consequences of them. You have myriad sensors which detect stretches and strains of structures inside the body. These deflections occur whether they are caused by a force or if they are caused by an acceleration term in the equations of motion, so they feel the same.

That's probably why we like to bundle them onto the force side of the equation so much. It "feels" to us like there's a force because the acceleration acts on these structures in our body in the same way a force does.

A person is sitting in a bus that is moving with constant velocity, due to friction their is no relative velocity between bus and the person.

If the bus is moving with constant velocity, no friction is required for the person to move along with the bus. However, even if the bus is accelerating no friction would be required anyway because the back of the person's seat will provide the necessary force to accelerate the person along with bus. The force applied by the seat back is $$f=ma$$ where $$a$$ is the acceleration of the bus and $$m$$ the mass of the person.

Only if the bus accelerates while the person is, say, standing on the bus and not holding onto anything will friction between the person's feet and the floor of the bus be required for acceleration of the person.

For a person standing outside, they observe that it isn't the person being moved backward but rather the bus accelerates to a new velocity and the person due to Newton's 1st law continues to move with their original velocity...

Again, the person will slide only if the maximum possible static friction force between a standing persons feet and the floor of the bus has been exceeded. But the person will not continue to move at constant velocity unless the friction force totally disappeared. That is not the case. Static friction becomes kinetic (sliding) friction which is generally lower than static friction. Kinetic friction acts forward on the person to oppose the backwards sliding motion. So the person will continue to accelerate but the acceleration will be due to the kinetic friction force and less that the acceleration due to static friction. (see below).

...until the normal force due to bus seat and the friction between them and seat accelerates them to new velocity of the bus.

It's not clear to me what you are saying, but the normal force $$N$$ applied to the person by the seat or floor depends only on the persons weight. So that force is constant whether the bus is accelerating or not. The maximum possible static friction force between the person's feet and the floor of the bus is

$$f_{max}=\mu_{s}N=\mu_{s}mg$$

where $$\mu_s$$ is the coefficient of static friction between the person's feet and the floor of the bus, $$m$$ the person's mass and $$g$$ the acceleration due to gravity. If it is not exceeded, then the static friction force applied to the person is

$$f=ma$$

where $$m$$ is the mass of the person and $$a$$ is the acceleration of the person and the bus. The person will start to lose traction with the floor when

$$f=f_{max}=\mu_{s}mg$$ $$ma=\mu_{s}mg$$ $$a=\mu_{s}g$$

So the person will start to slide back on the floor when the acceleration of person and bus equals $$\mu_{s}g$$. At that point the friction becomes kinetic (sliding). Now the acceleration of the person will be due to the kinetic friction force

$$a=\mu_{k}mg$$ where $$\mu_k$$ is the coefficient of kinetic (sliding) friction. Since, in general, $$\mu_{k}\lt\mu_{s}$$ the acceleration of the person will be less than the acceleration of the bus.