# Kinetic Energy derivation assumption?

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Roboto
The classical definition to the Kinetic Energy equation is KE=integral of F*dx where F=d(m*v)/dt. When mass is constant, KE=(1/2)m*v^2.

I am working on a vibration problem at work and having to review my Lagrangian Dynamics books from 30 years ago. So my question is about all of the authors of these books (and internet searches) make the same comment that this assumes that acceleration is constant. They make no further statement other than for simplicity they will use the classical definition for the Kinetic energy equation.

Why would they make that comment? When i use accelerations that are a function of time (and mass being constant) i can derive the same general equation form where kinetic energy is now just a function of time.

Though this is not a stumbling point for me, i am curious why all the authors would make that side comment that kinetic energy equation assumes that the acceleration being a constant.

Mentor
So my question is about all of the authors of these books (and internet searches) make the same comment that this assumes that acceleration is constant. They make no further statement other than for simplicity they will use the classical definition for the Kinetic energy equation.
Why would they make that comment? When i use accelerations that are a function of time (and mass being constant) i can derive the same general equation form where kinetic energy is now just a function of time.
Though this is not a stumbling point for me, i am curious why all the authors would make that side comment that kinetic energy equation assumes that the acceleration being a constant.
Because it's easier for the entry-point into the concept?

sophiecentaur
Gold Member
2022 Award
I don't understand the statement that the derivation of the "work-energy theorem" were only valid for constant acceleration. Where have you read this?

It's very easy to prove: Just define the kinetic energy
$$T=\frac{m}{2} \dot{\vec{x}}^2$$
and take its time derivative and use the equation of motion,
$$\dot{T}=m \dot{\vec{x}} \cdot \ddot{\vec{x}}=\dot{\vec{x}} \cdot \vec{F}.$$
If now ##\vec{F}## has a time-indepnendent potential,
$$\vec{F}=-\vec{\nabla} V(\vec{x})$$
you get
$$\dot{\vec{x}} \cdot \vec{F} = - \dot{\vec{x}} \cdot \vec{\nabla} V = -\mathrm{d}_t V(\vec{x}),$$
and then the work-energy theorem simplifies to
$$\dot{T}=-\dot{V} \; \Rightarrow \; \dot{E}=\mathrm{d}_t (T+V)=0 \; \Rightarrow \; E=\text{const},$$
i.e., energy conservation.

sophiecentaur
Gold Member
i.e., energy conservation.
In a conservative system, the Energy supplied must equal the Kinetic Energy out. However the Work is done, you have to get the same answer or we have a 'loophole' and the makings of a PPM.

Roboto
Yeah, I understand the conservation of energy, and the derivations. Easy peasy.

What I haven't figured out is why would my textbook authors, along with searches on the internet, have this comment that the derivation of the kinetic energy equation is based on the assumption that acceleration is constant.

So i thought i would bounce this off the group if they knew of cases where derivation kinetic energy would not be (1/2)m*v^2 if acceleration was not constant.

sophiecentaur
Gold Member
2022 Award
I've never read this nonsensical statement anywhere. There are of course some overly "didactical" high-school textbooks out there, which confuse the students more than helping them learning physics ;-).

sophiecentaur
Homework Helper
Gold Member
2022 Award
What I haven't figured out is why would my textbook authors, along with searches on the internet, have this comment that the derivation of the kinetic energy equation is based on the assumption that acceleration is constant.
When I googled "deriving kinetic energy equation", the top hit was:

https://physics.info/energy-kinetic/

Which does exactly as you say. As did the first video hit. Although the second video hit used more general acceleration.

Interesting. I suggest these are elementary introductions for those who haven't yet encountered non-constant acceleration.
So i thought i would bounce this off the group if they knew of cases where derivation kinetic energy would not be (1/2)m*v^2 if acceleration was not constant.
Well the relativistic kinetic energy is ##(\gamma -1)mc^2##, which is approxaimately ##\frac 1 2 mv^2## when ##v << c##.

Gold Member
I've never read this nonsensical statement anywhere. There are of course some overly "didactical" high-school textbooks out there, which confuse the students more than helping them learning physics ;-).
Yes: “confuse” is the word. Whether or not Mr Worthington actually said “constant acceleration”, he didn’t stress it. All the SUVAT equations he derived used straight lines (of course) but, despite him being a lovely old guy, he disturbed my sleep with that cognitive dissonance. All he need have said was “ assuming ……..”

vanhees71 and PeroK
Gold Member
2022 Award
When I googled "deriving kinetic energy equation", the top hit was:

https://physics.info/energy-kinetic/

Which does exactly as you say. As did the first video hit. Although the second video hit used more general acceleration.

Interesting. I suggest these are elementary introductions for those who haven't yet encountered non-constant acceleration.

Well the relativistic kinetic energy is ##(\gamma -1)mc^2##, which is approxaimately ##\frac 1 2 mv^2## when ##v << c##.
I don't think that it is a good strategy to use some movies or other arbitrary material from a Google search. Reliable are usually manuscripts accompanying lectures at universities and, of course, proper textbooks. If you insist on "movies" it's much less easy to give a general advice. You find some very good ones on youtube and in these "Covid times" also very many "online lectures", which also are usually good, but it is not surprising that on youtube there are also many bad ones. So, if you learn something new, so that you cannot easily judge yourself whether a source is reliable or not, it is important to cross-check with more reliable sources, i.e., lecture notes or, usually even better, textbooks and of course original scientific publications in peer-reviewed reliable journals. If the subject is not too new, you also find review articles as in journals like Review of Modern Physics or Physics Reports. For classical mechanics, of course, there are many good textbooks (the only exception is the >=3rd edition of Goldstein, which was spoiled by co-authors, while the 1st and 2nd editions are standard references).

Roboto
My Lagrangian Dynamics textbooks from 30+ years ago had the comment that the derivation of the classical Kinetic Energy equation assumes acceleration is a constant. I just don't understand why multiple authors would make comments that the derivation assumes acceleration is a constant.