# The math of physics in real world situations

My calculus I professor was an engineer, and liked to bring up that equations for events in the real world usually aren't pretty (non-linear).

With that in the back of my mind, I started my first physics course this semester and we are doing the basic one dimensional movement of a particle, position, velocity, acceleration, usually with a simple function of t.

I began to mesh the two ideas while I should have been paying attention, and it dawned on me, that in the real world (i know nothing of if this applies to actual particles, I mean bodies like a car or animal) no rate is ever constant.

For example, a car's velocity from a standstill to any speed is not going to be constant, it accelerates. It's rate of acceleration can't be constant either, and neither can it's rate of it's rate of acceleration, etc.

So, is it safe to say that a hypothetical equation that exactly modeled the position of the car could not be an equation that could be brought to a constant through repeated differentiation?

Just the thoughts of a first year physics major!

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So, is it safe to say that a hypothetical equation that exactly modeled the position of the car could not be an equation that could be brought to a constant through repeated differentiation?

Just the thoughts of a first year physics major!
How exactly would one actually construct such an equation ? You wold actually have to be able to do measurements with arbitrarily high precision. Physically you must always have some point where you end your equation.

K^2
Everything you do in physics has to do with models. Some models are better than others, usually at the expense of being more difficult to compute. The constant acceleration model for projectile, for example, is good to a very high precision at low velocities of projectile. If you need to get better results, you account for drag, and things get more complex.

The question is always what you need the results for. That tells you how precise you need them to be. After making computations, you can usually estimate error. If that error is too high to be acceptable, you go for a finer model or look for an alternative setup.

As far as linearity, most models are linearizable under certain conditions. Since linear models are the easiest for computations, that's often the route taken.

In reality, that equation you learned for constant acceleration doesn't seem to see much use outside of your physics class, gravity excluded. For your car example, you just need to think of the calculus relations. If you have a function that describe acceleration, integrating with respect to time gives velocity. Integrating again gives position. So maybe your acceleration function is polynomial, and maybe it's not. And of course it only has a certain range of validity (domain), so in reality, it all depends on what your acceleration function is.

To go into more depth, perhaps we know the power curve for the car's engine. And then the forces acting on the car such as drag. Lets say that the engine can go up to a certain rpm, and then there's gearing for the wheels so they turn proportional to that. The acceleration is then a function of the change in kinetic energy, where that is power left after you've fought drag. And when we get to a certain point where the engine can't go any faster, we shift gears and repeat, so the acceleration function is always changing.

BruceW
Homework Helper
So, is it safe to say that a hypothetical equation that exactly modeled the position of the car could not be an equation that could be brought to a constant through repeated differentiation?

Just the thoughts of a first year physics major!
You're right that a general equation can't be brought to a constant with repeated differentiation.

Luckily for us, there are some situations where we can say the acceleration is approximately constant, for example the path of a ball acted on by gravity.
And there are situations where velocity is almost constant, i.e. a car travelling at almost constant speed on the motorway.

phinds
Gold Member
2019 Award
Everything you do in physics has to do with models. Some models are better than others, usually at the expense of being more difficult to compute. The constant acceleration model for projectile, for example, is good to a very high precision at low velocities of projectile. If you need to get better results, you account for drag, and things get more complex.

The question is always what you need the results for. That tells you how precise you need them to be. After making computations, you can usually estimate error. If that error is too high to be acceptable, you go for a finer model or look for an alternative setup.

As far as linearity, most models are linearizable under certain conditions. Since linear models are the easiest for computations, that's often the route taken.
What a nicely stated, concise, accurate answer. Wish I could write like that (I get too long-winded).

One way in which a complex model can be reduced to linear one is by a taylor series expansion.