Can the equations of motion be used for both instantaneous and average quantities?

Question

I have three equations of motion: $$v = u+ at , $$$$s = ut + \dfrac{1}{2} at^2 , $$$$ v^2 = u^2 + 2as.$$ Can I use them for both finding instantaneous and average quantities? Suppose I want to find the final instantaneous velocity then can I use the first equation; or if I want to find average velocity over a time period, can I use the first equation?

Answer 1

Well, the three equations that you mentioned are all derived from the definitions of velocity and of acceleration. I'm assuming you know calculus (if you don't, just ignore the next part and look at the conclusions, or even better- learn calculus!). $$v=\frac{dx}{dt}$$ $$a=\frac{dv}{dt}$$

Now, to find displacement we just look at the first equation: $$\Delta{x}=\int v dt$$

But velocity as a function of time is $$v=\int a dt+v_0$$

So displacement will be $$\Delta{x}=\int \left(\left(\int a dt\right)+v_0\right)dt$$

And when the acceleration is constant, we can say that this is $$\Delta{x}=\int \left(at+v_0\right)dt=v_0t+\frac{1}{2}at^2$$

If you want to find the instantaneous velocity, then you need to use the first equation I listed, which defines instantaneous velocity. However, when you have constant acceleration, the first equation you mentioned works, since with constant acceleration we have (from my fourth equation), $$v=\int a dt +v_0=at+v_0$$

Conclusions

You can use all the formulas you mentioned, but only when you have situations with constant acceleration. I showed you that those equations of motion are equivalent to the definitions of velocity and acceleration, but only if you have constant acceleration. In that case, then you can find instantaneous speed comfortably, as well as the position of a particle at any point in time.

The third formula which you gave, I haven't mentioned yet. That one's derived through some straightforward algebra from the other two you listed, so I'll let you do that on your own if you haven't yet; given that it's derived from the other two, the third equation also works only in cases with constant acceleration (it can also be derived from the work-energy theorem, that's something you could have fun with later on if you haven't seen it yet).

As for average quantities- if you wanted average velocity, I guess you could say that $$v_{avg}=\frac{\Delta{x}}{\Delta{t}}$$ Which refers to the total displacement as opposed to the infinitesimal one above.

One subtlety with regards to average velocity that we should be careful about: average velocity could refer to the definition I gave you above, which is just displacement over time, or it could refer to mean velocity, which is the sum of the instantaneous velocities divided by the number of instantaneous velocities. I assumed you meant the first definition of average velocity.

If you wanted to find average velocity over a time period, you couldn't really use the first equation you mentioned. As I said, that gives you the instantaneous velocity. (In constant velocity motion, instantaneous velocity=average velocity so I guess the equation works there but that's a very special case.)

So essentially, those equations work if you want to find the final instantaneous velocity or the displacement, but they're not really helpful for finding things like average velocity. Just keep in mind that the acceleration has to be constant.

This is my first post on this site, hopefully I was able to help you out!

Ps. I hope you don't mind that I switched the names of some variables. I find it more comfortable to name the initial velocity $v_0$ and the displacement $\Delta{x}$.