# Does the force in a car going over a bump depend on the car's mass?

Imagine 2 cars 1 weighs 1 ton, the other weights 2 tons. If we ignore the effect of the suspension - when both of these cars go over the same bump in the road, which accelerates fastest in the vertical direction?

I would have assumed that the lighter car would - as $a=F/m$, so the car with more mass has a lower acceleration. But inversely - would the force input to both cars be the same? Or would this be mass dependent and both cars would accelerate at the same rate?

I guess in simple terms, I'm asking is the force input constant, or is the acceleration constant between the 2 cars - or something in between?

• Why would they accelerate at different rates? Is acceleration in the vertical direction really what you want to ask about? What are you trying to quantify? – QtizedQ Aug 3 '17 at 14:24
• What is your motivation for ignoring suspension? – JMac Aug 3 '17 at 14:56
• Hi QitzedQ, I would have assumed they would have accelerated at different rates due to the difference in mass. I.e same amount of force input to the cars wheels = varying acceleration due to the mass delta? – Jonny Plutonium Aug 4 '17 at 9:32
• They (whatever you mean by "they" - the 4 different wheels?) will accelerate differently because they are not all hitting the same pothole at the same instant in time. But on its own, that fact doesn't tell you anything very useful. – alephzero Aug 4 '17 at 14:08
• They = the two cars of differing masses. – Jonny Plutonium Aug 4 '17 at 18:35

If you ignore the suspension (which is ignoring everything that actually matters) then it makes no difference. As the car goes over the bump it converts kinetic energy it has by virtue of going along the road to vertical kinetic energy and then to potential energy at the top of the bump, and kinetic & potential energy are both proportional to mass, so mass can be factored out and position and all its differentials with respect to time -- in particular velocity and acceleration -- is the same.

But, as I said, don't ignore suspension (or, equivalently, try driving a car with no suspension for a bit and you will quickly understand why you can't ignore it!)

• This beautifully illustrates the difference between physics and engineering. Engineers ignore things that don't matter in real life. Physicists ignore things so that the problem is easier to solve - and then design lab experiments that don't include those things, to verify the solution is correct! – alephzero Aug 4 '17 at 0:04
• @alephzero: I think it is specific to some kinds of physics -- I work with climate people, who I think really are physicists at heart, and one of the things I've discovered is that my trick of 'let's ignore the boring complexities' gets the wrong answers: the system is actually irreducibly complicated and you can't do the sort of tricks that my sort of physicist do at all. This has been an interesting lesson to me. – tfb Aug 4 '17 at 6:29
• Unclear. When you say everything else is the same do you mean that the force is the same or the acceleration is the same? I think the acceleration is the same, therefore the force will be different. – sammy gerbil Aug 5 '17 at 9:51
• I meant the velocity and all its differentials (and specifically not the forces): sorry. I will amend the answer. – tfb Aug 5 '17 at 19:15

Like tfb said, once you factor out the mass the acceleration is the same. When you factor the mass back in the force on the heavier car is greater.

If you think about both cars traveling the same path over the same amount of time, the kinematic descriptions of both cars are the same. Newton's second law gives the force proportional to mass with the same acceleration.

Also consider the bump to be of infinitesimally small height. The normal force from the ground is higher for the heavier car, therefore the bump exerts more force on it.

Lastly, consider a bump made of cardboard which is capable of exerting some small force. A light car could drive over it, but a heavier car would collapse it. That is because the heavier car requires more force upwards for it to drive over the bump.