# Fictitious Forces in a car crash

Do we actually feel fictitious forces acting on our bodies? My conclusion is that yes we do, since when I am accelerating upwards in an elevator, I feel a lot of pressure on my knees. I have an example though (that certainly has some logical error in it) that proves the opposite:

A car is moving at constant velocity u=100km/h with a passenger on the back seat. The passenger is sitting on ice so that the friction between the car and passenger is 0. The car crashes on a wall, and stops completely in t=0.1s.

Inertial frame of reference:

An observer outside of the car, sees the passenger moving at constant speed of u=100km/h after the crash, so the net force acting on him continues to be 0. When the passenger eventually hits the wall he stops moving as well.

Non inertial frame of reference:

There is a camera placed on the window of the back seat focusing on the passenger. As the car crashes, the people watching from the camera see the passenger accelerating from 0 to 100km/h relative to the camera. Since there is no other way to explain this, they assume that a force acted on the passenger, causing his acceleration. Since the passenger accelerated 100 km/h in 0.1 seconds we have a great rate of change of momentum, causing an enormous force.

Conclusions:

1. In the inertial frame of reference we expect the passenger to be damaged after hitting the wall.

2. In the non inertial frame of reference we expect the passenger to be damaged instantly, due to the force acting on him.

The second conclusion can't possibly make any sense, cause if it did I could hurt people by moving next to them very quickly. What's the difference between the fictitious force in an elevator and the one in the crashing car?

• What is your definition of a 'fictitious force'? – Time4Tea Aug 23 '18 at 17:21
• A force that explains acceleration in a non inertial frame of reference, but doesn't really exist. For example if I leave a weightless ball next to me inside of an accelerating elevator, the ball starts accelerating downwards for no apparent reason. So I assign a force to it to explain that acceleration. The force doesn't really exist though since someone standing outside of the elevator sees the ball standing still as it should, with no forces acting on it. Did I understand something wrong? – Dimitris Pozoukidis Aug 23 '18 at 17:32
• If the ball is weightless (massless), then it doesn't require a force acting on it to accelerate. $F=ma$ – Time4Tea Aug 23 '18 at 17:41
• Alright then bad example. The ball has a mass and it should accelerate with g downwards due to its weight. When I leave it though I see it accelerating much faster than that. I can explain that only by saying that a force other than its weight acts on it. That is my fictitious force, since the observer outside of the elevator sees just the weight of the ball. – Dimitris Pozoukidis Aug 23 '18 at 17:56

Fictitious forces are introduced in a non-inertial frame of reference so that we can use Newton's laws of motion.

Do we actually feel fictitious forces acting on our bodies? My conclusion is that yes we do, since when I am accelerating upwards in an elevator, I feel a lot of pressure on my knees.

Suppose that an elevator is accelerating upwards with an acceleration $a$ relative to the ground.
You of mass $m$ are standing in the elevator and have two forces acting on you.
The upward force on you due to the floor of the elevator $N$ and the downward attractive force on you due to the Earth $mg$.

Considering you as the system and applying Newton's second law gives $N-mg = ma$ if up is taken as the positive direction.

So $N = m(g+a)$ and that is the force that you feel pushing up from the ground.
You might think of it as you having increased your weight and your knees are feeling the effect of you increased "weight".
$N$ and $mg$ are real forces.

Now move to the non-inertial frame of the accelerating elevator.
You are not moving relative to the elevator and yet there is a net upward force on you, $N-mg=ma$, which is not good news if you want to use Newton's laws of motion.
You remove the anomaly by applying a fictitious downward force of $ma$ on you so that now the net force on you is zero, $ma + (-ma) = 0$, which is consistent with you being at rest relative to the elevator.

An observer outside of the car, sees the passenger moving at constant speed of u=100km/h after the crash, so the net force acting on him continues to be 0. When the passenger eventually hits the wall he stops moving as well.

Let the acceleration of the car relative to the ground be $-a$ ie in the opposite direction to the initial velocity of the car.

Move to the non inertial frame of reference - the car.
Relative to the car you see the passenger, mass $m$ accelerating at $+a$ in the direction of the original velocity of the car.
Again there is a problem as the passenger has no force acting on her and yet she is accelerating - contrary to Newton's laws of motion.
How do we make Newton's laws work.
Apply a fictitious force of $+ma$ on the passenger.
This force does not kill the passenger as it does not exist, the fictitious force is there to enable us to use Newton's laws of motion in a non-inertial frame.
When the car stops moving the car now becomes an inertial frame of reference and the passenger is observer to fly though the air at constant velocity with no horizontal forces acting on her.
When the passenger hits the wall the real force on the passenger due to the wall does damage to the passenger.

I am accelerating upwards in an elevator, I feel a lot of pressure on my knees.

How is that fictitious? The floor is moving up against you, making you the non-moving object into a moving object, and you feel that force until such time as you moving the same speed as the floor.

when I am accelerating upwards in an elevator, I feel a lot of pressure on my knees.

You use this as an example of feeling fictitious forces. But the floor is exerting real forces upward on you. Your example doesn't separate them.

Do you "feel" gravity? Not really.

What your body "feels" is the fact that forces are applied to one portion of your body (perhaps your feet) and not to others. If you could apply a uniform force to every portion of your body simultaneously, you wouldn't notice. This is true of gravity and other fictitious forces.

What's the difference between the fictitious force in an elevator and the one in the crashing car?

In the car, you're starting in the middle (away from the "wall"). In the elevator, you're already standing. In both cases your body can't feel the fictitious force, but it feels the pressure from the container as it changes your acceleration (the floor of the elevator, or the front of the passenger compartment).

The main difference in the two situations is just our interpretation. We are used to thinking of the forces from the floor (and the other internal forces in our bodies) as "weight" or gravity pulling down. But really what you're "feeling" is the floor pushing on your feet (and your shoulders pulling on your arms, etc.) If we open a trap door under you, gravity is still present, but you don't feel heavy any more.

If you can get around to that point of view, the car works exactly the same way.