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Can anyone explain how force is exerted on car seats during a crash?

I am being told in car seat safety forums that $force=mass \times acceleration$. So if a $2000 \;\text{lb}$ vehicle bumps you at even $5 \;\text{mph}$, the force exerted on your car seat equals $10000 \;\text{lb}$, which is good reason to replace the seat.

My husband (an engineer), says it's not that simple and that the mass being used in the above equation should not be the mass of the vehicle but that of the child and car seat combined. So at $5 \;\text{mph}$ the force exerted on the car seat would be something much less, on the order $500 \;\text{lb}$, depending on total weight of the seat and child. He also said something about the energy transfer into the brakes, the bumper, and car seats - therefore not all of that force is being applied to the car seat.

What mass should be used to calculate the force exerted on a car seat during a collision, $m_\text{vehicle}$ or $m_\text{seat+child}$?

Can anyone shed some light on this or explain better what either my husband or the car seat safety forums are trying to say?

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  • $\begingroup$ What gets accelerated by the seat is the child, so it's the mass of the child that applies. The car accelerates the seat plus the child, of course, but it's more than likely that the frame of the car can take that. At 5mph even the car seat and the body of the child should be able to absorb the forces that accelerate the child without damage, but I wouldn't push it much beyond that... $\endgroup$
    – CuriousOne
    Nov 2, 2015 at 1:57
  • $\begingroup$ Sorry, none of this makes any difference. The physics of whether or not your car seat is damaged is not something that you're going to be able to determine with a simple application of Newton's 2nd law. The overall amount of force is only one variable that will affect how damaged or not your car seat is. You should follow the manufacturer's guidelines - they're the ones who have done crash tests with your make & model seat to determine when it's safe and when it isn't. $\endgroup$
    – Brionius
    Nov 2, 2015 at 2:17
  • $\begingroup$ @Bronius: She asked if we can explain to her what forces are being exerted on the car seat. Yes, we can explain that. Whether the car seat is safe or not depends on a lot more than on the guidelines. If the manufacturer is not honest (and that wouldn't be the first time child car seats are under fire), then those guidelines are meaningless. Irrespective of that the seat only has to hold the child, not the car. $\endgroup$
    – CuriousOne
    Nov 2, 2015 at 2:41
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    $\begingroup$ I really hope that "I am being told in car seat safety forums that force=massXacceleration, so if a 2000lb vehicle bumps you at even 5mph you are looking at 10,000lb of force exerted on your car seat" represents some loss of information along a whisper chain, rather than something that someone serious is actually putting out there. Sheesh, the units don't match up. But, no matter. Your husband is right that this math is badly bogus. $\endgroup$ Nov 2, 2015 at 3:03

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The short answer- The force exerted on a car seat depends on how much your car crushes to absorb the impact. The more your car deforms, the lower the force.


When you are thinking about this type of problem, remember that $5 \text{mph}$ is a speed (velocity), not an acceleration. Acceleration describes how quickly an object changes speed ($a = \frac {v_{f}-v_{i}}{t}$). As you correctly noted, $F = ma$, however, forces are caused by accelerations, not speeds!

**To calculate Force in [$\text{lbf}$]: Acceleration has units [$\frac {ft}{s^2}$], with velocity in [$\frac {ft}{s}$] and time in [$\text{s}$]; Mass has units [$\text{slug} = \frac {\text{lbm}}{\text{g}}$] where $g=32.2 \;[\frac {ft}{s^2}]$.

Calculate Acceleration:

$$a = \frac {0 \;\text{[ft/s]} - 7.33 \;\text{[ft/s]}}{t}$$

This is where your question becomes a matter of guess work, because we dont know how long the crash takes- it depends on how much your car deforms during a crash. As a ballpark guess, lets assume that the crash takes $t = 0.25 \;\text{seconds}$.

$$\therefore a = \frac {0 \;\text{[ft/s]} - 7.33 \;\text{[ft/s]}}{\approx 0.25 \;[\text{s}]} \approx -29 \;[\frac {ft}{s^2}]$$

Calculate Force:

The mass of the driver and the mass of the car seat are accelerated with the car body. Assuming the driver weighs $150 \;\text{[lbm]}$ and the car seat weighs $50 \;\text{[lbm]}$, their masses are $4.6 \;\text{[slug]}$ and $1.5 \;\text{[slug]}$. The force on the car seat is calculated by:

$$F_{seat} = (m_{driver}+m_{seat})a \qquad \Rightarrow \qquad \therefore F_{seat} \approx 182 \;\text{[lbf]}$$

Similarly, to determine force exerted on the driver:

$$F_{driver} = (m_{driver})a \qquad \Rightarrow \qquad \therefore F_{driver} \approx 137 \;\text{[lbf]}$$

In reality, the seat cushion compresses, increasing the impact time (felt by the driver). This decreases the drivers acceleration and reduces the force felt by the driver.

While braking does exert force on the seat and driver, it is calculated separately because it occurs before the collision. For example, you might calculate the force as the car decelerates from $30 \;\text{[mph]}$ to $5 \;\text{[mph]}$ (in some time interval) before the collision. However, this deceleration is much less than the deceleration/force involved in the collision and can be safely ignored. After all, you dont need to replace a car seat after braking!

The car seat is easily strong enough to handle these forces. However, this is a concern in high speed crashes, where forces are large enough to create stresses in the seat frame that exceed the yield limit, the seat should be replaced.

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Can anyone explain how force is exerted on car seats during a crash?

I am being told in car seat safety forums that force=mass×acceleration. So if a 2000lb vehicle bumps you at even 5mph, the force exerted on your car seat equals 10000lb, which is good reason to replace the seat.

The formula is correct. $F=ma$. This is the total force $F$ it takes to accelerate a mass $m$ with the acceleration $a$.

Now, the point is - maybe surprisingly - not the speed of the car! The point is not how fast you are going (or how fast the bumping car is going).

The point is how fast you are decelerating. In other words, how fast you are speeding down! If you drive the car strait into a stone wall, it is a lot more damaging for the child on the back seat than if you drive into a large soft vertical foam matress.

Why? Because the foam matress makes the car (and you) slow down slowlier! It simply takes longer for the speed to go from 5mhp to 0mhp, which means that the deceleration (negative acceleration) $a$ is smaller and therefore the force $F$ is smaller as well.

My husband (an engineer), says it's not that simple and that the mass being used in the above equation should not be the mass of the vehicle but that of the child and car seat combined.

Yes, because $F$ in the formula above is the total force needed to accelerate the mass $m$. And what you are interested in is the force on the child, so the child would be your $m$. The force on the car might be very different and much larger than the force on the child.

I am not quite sure from the question, exactly what object you wish to look at, so I've assumed we are looking at the impact on the child. If you wish to look at the seat alone, we should define what the seat exactly is (or is a childs seat?)

He also said something about the energy transfer into the brakes, the bumper, and car seats - therefore not all of that force is being applied to the car seat.

Yes, here comes a point of car evolution and improvements. This is the reason that we cannot just calculate the impact force on people in a car - it is way too complicated with way too many unknown factors - so crash tests are what is being used.

The point is again that the foam wall is making the impact softer because is gives the car more time to slow down from the 5mhp to 0mhp. We want to avoid a sudden stop, which would cause a large deceleration.

So, if we do hit a stone wall, we must instead imitate the foam somewhere else. For example in the structure of the car before the impact reaches the child on the backseat.

In accidents you very often see extremely damaged and crumbled cars - it looks very violent even when the people inside weren't that injured. That is because many bigger cars are made so their frontend will crumble a lot. This is like a foam sponge that absorbs the energy because it slows the car down more slowly than if the car was totally rigid.
This foam sponge effect is the case in many cases throughout the car chasis - everywhere where the car is not perfectly rigid, energy is absorbed for the car deformation giving the rest of the car and passengers more time to slow down.

The airbag on the frontseats works like a pillow that your head can hit. Then, when the head flies forward, because the car is being stopped suddenly, it is slowly slowed down by this "pillow" and decelerated less just over a longer period of time. If the airbag wasn't there, the head might continue with the same speed until it hits the dashboard and there it would be stopped suddenly and experience a large deceleration. Or, the neck would have to stop the head and maybe the bones and the body is not strong enough the cause this deceleration of the head, so something will break.

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I am not a physicist

The force applied to the seat will be the net (difference) of all the following:

  1. Force absorbed by the car body.
  2. Force absorbed by the brakes/friction.
  3. Force dissipated through the frame (on to which the car seat is attached).
  4. Force absorbed by the car seat itself.

Finally, whatever may be left will be transferred to the occupant.

The only way you can affect a 10,000 lbs of force onto an object is if you discount all the other forces in effect and exert a much larger force in terms of the momentum of the object.

Keep in mind that vehicles are designed with a certain amount of safety and that there are many mechanisms in place (in terms of the structural design) absorb any force and dissipate it away from the occupants. These design mechanisms include the placements and size/shape of the seats.

At 5 miles per hour - which is nearly idle speed any bump to a vehicle will result in at most a small dent to the part of the body hit by the vehicle; the force exerted would not be sufficient to structurally compromise the car seat such that it would need replacement.

Source: Sadly, experience with car accidents first hand.

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Your husband is right. The mass to use in that equation is the mass of the car seat and passenger combined.

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