Understanding the 1-D Kinematics Equations

[gibberingmouther]

So, the equations I'm talking about are the "big 4" listed here: https://www.physicsclassroom.com/class/1DKin/Lesson-6/Kinematic-Equations

I understand how to derive all these using calculus or algebra and graphs. That's not my problem. I can apply them pretty well to problems I need to solve as well.

I want to go a step further and understand *why* these equations work using my regular thinking processes, and not just kind of see how to step by step derive them. It's hard to explain what I mean.

For example, I *do* feel I "understand" the equation: vf=vi+a*t

The initial velocity is there because of inertia, and the acceleration is constant so knowing that and the time interval will give you how much the object has accelerated by. Thus, adding these, you get the final velocity. I feel I really *get* this equation and understand why it is true.

I do not feel the same way at all about the remaining three equations. Like I try to understand how these apply to an object moving at a constant acceleration, and I got nothin'. Can anyone help with this? Is it feasible to try and think like this as you go further in learning physics equations?

EDIT: Okay, I found a good explanation for the x=vi*t+.5*a*t^2 equation: https://www.physicsclassroom.com/class/1DKin/Lesson-6/Kinematic-Equations-and-Graphs
So I'm good with 2/4 of them now!

EDIT2: Okay, I got the one for average velocity times the time interval too now. Learned a bit working on that one, such as about what it means to average things (including the integral definition for the average value of a function over an interval). So, one more, I'll resume this tomorrow. Perhaps the more important remaining question is whether this sort of approach will work with more advanced topics? So, for example, the "understanding" of the displacement equations involved graphs. The understanding for the vf equation involved knowing about the law of inertia and just the definition of acceleration. A combination of working things out on my own and Googling.

 

[Delta2]

If I understand correctly your post, you seek out for an intuitive understanding of the various equations you learn in physics. Although this is probably possible, and certainly would be a plus to always have an intuitive understanding, still this is not my personal approach. I just have an intuitive understanding of some equations and concepts I consider to be central and basic and fundamental. And the rest of equations I consider them to be a consequence of the basic equations via mathematical processing using algebra or calculus or other mathematical techniques.

For example for this case of the big 4 equations I don't have myself a particular intuitive understanding of them (though it would be good and desirable to have an intuitive understanding of them since one can consider those 4 as basic and fundamental). I just have an intuitive understanding of the key equations which are $v=\frac{ds}{dt}$ and $a=\frac{dv}{dt}$ and I know I can derive the "big 4 equations" as a consequence of those two basic equations, via the mathematical processing of integration (which I have to say I have both an intuitive and formal understanding of why integration is the reverse process of differentiation) and some algebraic processing.

 

[vanhees71]

Hm, I strongly suggest to find a better source to learn mechanics, e.g., a good introductory textbook like Tipler.

 

[A.T.]

Perhaps the more important remaining question is whether this sort of approach will work with more advanced topics? So, for example, the "understanding" of the displacement equations involved graphs.

Check out the below video series for a more intutive understand of calculus:

 

[Delta2]

Hm, I strongly suggest to find a better source to learn mechanics, e.g., a good introductory textbook like Tipler.

Yes that would be better to do BUT no matter how good a book is it can't provide an intuitive treatment of each and every equation it derives.

For example take this equation from the "big 4" $V_f^2=V_i^2+2ad$ and you are asked "What's the intuition behind it?". I don't know if you can imagine some good intuition involving Pythagorean theorem (because we have two quantities squared, don't know I am just saying) or some other intuitive explanation. For me I don't see any particular intuition behind it, I just see it as the result of typical algebraic processing of the other two equations $V_f=V_i+at$ and $d=V_it+\frac{1}{2}at^2$ which two can be derived by integration for the case of motion with constant acceleration $a$.

EDIT: OK it turns out that there is an intuitive understanding of this equation. If we multiply both sides of the equation by the mass m of the particle and divide by 2, we get that the final kinetic energy of the particle equals the initial kinetic energy plus the work done.

 

[vanhees71]

I have no clue what "the big 4" might be. Physics is not about learning some equations by heart but to understand concepts. Obviously you are talking about motion in one direction with a constant force acting in this direction on a point particle. This implies the equation of motion
$$m \ddot{x}=F=\text{const},$$
from which everything can be derived. For a physicist it's more important to know the methods to derive the properties than learning them by heart.

 

[CWatters]

The image presumably with the 4 equations doesn't appear for me.

Vf^2 = Vi^2 + 2as

Comes from conservation of energy.

Try multiplying both sides by 0.5m and you get final KE equals initial KE plus work done.

 

[Serious Max]

The image presumably with the 4 equations doesn't appear for me.

Vf^2 = Vi^2 + 2as

Comes from conservation of energy.

Try multiplying both sides by 0.5m and you get final KE equals initial KE plus work done.

Interestingly, Shankar (Yale) in his YouTube lectures actually derives it the other way around: assumes we know (as he derives in an earlier lecture) this kinematic formula and derives the formula for work. The kinematic formula was derived from combining other two.

(10:30; can't link)

 

[gibberingmouther]

Okay, I basically used one or two websites and by looking at graphs I was able to develop a good enough intuition for the 4 constant acceleration 1-D kinematics equations.

The connection to conservation of energy was also really interesting!

And I had to think hard about what an "average" is. I found the definition for the average value of a function in terms of the average value integral. That was cool.

I started working on E&M mechanics out of boredom (I plan to take this class next semester so I figure it can't hurt to start now). So now I have some experience with other equations describing a different part of the phenomena of nature ... seems like "intuition" is still a possibility here!