Does it make sense to model the kinetic energy as $T=\frac{1}{2}mv\,|v|$ instead of $T=\frac{1}{2}mv^2$?

Question

Does it make sense to model the kinetic energy as:

$$T=\frac{1}{2}mv\,|v| \ \ \ \text{instead} \ \text{of} \ \ \ T=\frac{1}{2}mv^2 \ \ \ ?$$

In the following youtube video, it is explained for an airfoil, why the classic Bernoulli Law description is not enough to explain how much lift is generated in reality. I am not interested in the topic of the video "per se", but in the construction of his explanation. The author uses the Bernoulli Equation to group the terms associated with the kinetic energy of the fluid. After some manipulation, using it to built the Drag Force proportional to a quadratic term of the velocity, giving intuition of why it is modeled as:

$$F_d=\frac{1}{2}\rho A\,C_d\,v^2$$

Now, I would like to review a simple and well known example of a physics model, where the drag force is considered, the classic nonlinear pendulum with friction.

If the classic nonlinear pendulum with the friction equation is reviewed, where the drag force is modeled proportional to the speed, as in Stokes' Law, in Wolfram-Alpha. It can be seen that it has decaying solutions as expected:

$$\ddot{x}+2\cdot0.021\,\dot{x}+0.2\sin(x)=0, \ \ \ \\ x(0)=\frac{\pi}{2}, \ \ \ \ \dot{x}(0)=0 \tag{Eq. 1}\label{Eq. 1}$$

If instead, the standard drag force $F_{\text{drag}}\propto (\dot{x})^2$ is used as is shown here for the equation:

$$\ddot{x}+0.021(\dot{x})^2+0.2\sin(x)=0, \ \ \ \ \\ x(0)=\frac{\pi}{2}, \ \ \ \ \ \dot{x}(0)=0 \tag{Eq. 2}\label{Eq. 2}$$

their solution isn't showing the expected decay one can see on the experimental pendulums.

This issue could be solved using an ansatz for the drag force $F_{\text{drag}}\propto \dot{x}|\dot{x}|$ (following this reference equation $1.127$), which as can be seen here for the equation:

$$\ddot{x}+0.021\dot{x}|\dot{x}|+0.2\sin(x)=0, \ \ \ \ \\ x(0)=\frac{\pi}{2}, \ \ \ \ \ \dot{x}(0)=0 \tag{Eq. 3}\label{Eq. 3}$$

their solution has recovered again the expected decaying behavior for a pendulum with friction.

Now, given the close relation between the kinetic energy and the drag force shown on the video, and due the failing of the classic form of the drag force to reproduce the decaying solutions of a pendulum with friction, I would like to know if make sense to consider the kinetic energy as:

$$T=\frac{1}{2}mv\,|v| \ \ \ \text{instead} \ \text{of} \ \ \ T=\frac{1}{2}mv^2$$

I hope you could explain why that is so, and if there are any examples, sharing them and explaining how the kinetic energy term $T\propto v\,|v|$ has arisen. For a negative answer, please elaborate, explaining why the standard equation of the drag force shown in Wikipedia fails at describing the pendulum with friction as shown in \eqref{Eq. 2} but it works for \eqref{Eq. 3} (in this paper at point $III$ is even solved piecewise).

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Added Later

These are the calculation done in the mentioned youtube video:

$$\begin{array}{l} \text{velocities on upper/lower sides of the airfoil}\quad v_1=\frac{d_1}{t_1}; \quad v_2=\frac{d_2}{t_2}; \\ \text{assumption}\quad t_1=t_2\ \ \textit{(mistaken)}\\ \text{Bernoulli Equation} \qquad P_1 + \rho g h + \underbrace{\frac{1}{2}\rho v_1^2}_{\text{kinetic energy}} = P_2 + \rho g h + \underbrace{\frac{1}{2}\rho v_2^2}_{\text{kinetic energy}} \\ \Rightarrow \Delta P = \frac{1}{2}\rho\left(v_2^2-v_1^2\right) = \frac{1}{2}\rho\left(\left(\frac{v_2+v_1}{v_1}\right)\left(\frac{v_2-v_1}{v_1}\right)\right)v_1^2 = \frac{1}{2}\rho\left(\left(\frac{v_2}{v_1}+1\right)\left(\frac{v_2}{v_1}-1\right)\right)v_1^2 \\ \Rightarrow \Delta P = \frac{1}{2}\rho\left(\left(\frac{d_2}{d_1}+1\right)\left(\frac{d_2}{d_1}-1\right)\right)v_1^2 = \frac{1}{2}\rho\left(\left(\frac{d_2}{d_1}\right)^2-1\right)v_1^2 \\ \text{multiplying both sides by area} \Rightarrow \underbrace{A\Delta P}_{F_d} = \frac{1}{2}\rho A \left(\left(\frac{d_2}{d_1}\right)^2-1\right) v_1^2 \underbrace{\propto}_{\text{proportional}} \frac{1}{2}\rho A\,C_d\,v_1^2 \end{array}$$

since from the kinetic energy part of the Bernoulli Equation the video find the Drag Force for the example, I made the pairing with the version were the absolute value is used. Hope this better explain why the question arises.

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2nd added later

I found this paper:

• "The pendulum—Rich physics from a simple system" - Robert A. Nelson & M. G. Olsson

where the autors on equations $(59)$ and $(60)$ introduce a Drag Force $F_d = b\ \dot{y} + c\ \dot{y}|\dot{y}|$ for a more accurate description of the air effects on the pendulum. I don't know if it add info about the Kinetic Energy question, but it make sense of the signs required for the proper description of the drag force.

Answer 1

No, because kinetic energy is not a vector, so we cannot have it be proportional to a vector. Kinetic energy is a scalar quantity, and so $T=\frac12m\mathbf v\cdot|v|$ cannot be used. The drag force is a vector, and it's related to the direction of $\mathbf v$, so this is why we have the expression you give in your post.

In general, this shows issues with questions like these asking "why can't we use this expression for this physical quantity instead?" The problem is that you aren't linking this question to any other definition of kinetic energy, and so the question misses the purpose of definitions. "Kinetic energy" is the quantity whose change is equal to the net work done. From this, we find $T=\frac12mv^2$ is the expression that describes this definition.

If you have another understanding of kinetic energy, you should ask why $T=\frac12m\mathbf v\cdot|v|$ can't be used to describe the definition you have in mind. Otherwise there isn't much else to be said.

Answer 2

1. The drag force $${\bf F}_{\rm drag}~=~-f(v^2)~ {\bf v}, \qquad f(v^2)~>~0, \tag{1} $$ is always directed opposite of the velocity ${\bf v}$:

   1. Linear friction/drag corresponds to a constant $f$-function.

   2. The $f$-function is a square root for quadratic drag. In 1D this becomes an absolute value, cf. OP's post.

2. It is unclear what a quadratic drag force ${\bf F}_{\rm drag}\propto-\dot{x}|\dot{x}|$ has to do with a signed kinetic energy $\frac{1}{2}m\dot{x}|\dot{x}|$ in 1D, cf. OP's post.

   Nevertheless, a signed kinetic energy is e.g. useful in the 1D double-integrator, cf. this Phys.SE post.

Answer 3

Drag force
The reason for failure of the drag force equation, is that the drag force should be directed against the velocity, $\dot{x}$. The Stoke's term, linear in velocity (as well as any odd power of velocity) satisfies this condition $$ F_{2n+1}(\dot{x})=-\gamma\dot{x}^{2n+1}=-F_{2n+1}(-\dot{x}). $$ On the other hand, an even power of $\dot{x}^2$ requires a bit more involved math, as, e.g., $$ F_{2n}(\dot{x})=-\gamma\text{sign}(\dot{x})\dot{x}^{2n}= =-\gamma\dot{x}|\dot{x}|^{2n-1}=-F_{2n}(-\dot{x}). $$

Energy
The reasons that the energy is a scalar are quite different - it is the first integral of the equations of motion. E.g., if we take Newton's equation for a particle in a conservative field $$ m\ddot{\mathbf{r}}=-\nabla U(\mathbf{r}), $$ multiply it by $\dot{\mathbf{r}}$ and perform some algebraic transformations, we obtain: $$ 0=\left[m\ddot{\mathbf{r}} +\nabla U(\mathbf{r})\right]\dot{\mathbf{r}}= m\ddot{\mathbf{r}}\dot{\mathbf{r}} +\nabla U(\mathbf{r})\dot{\mathbf{r}}= \frac{d}{dt}\left[\frac{m\dot{\mathbf{r}}^2}{2}+U(\mathbf{r})\right]=\frac{d}{dt}E(t) $$ That is, the quantity $E(t)=\frac{m\dot{\mathbf{r}}^2}{2}+U(\mathbf{r})$ does not change with time (i.e., conserved) along the trajectories described by the equation of motion. This is just a mathematical fact - there is no much flexibility in adjusting it.

Answer 4

Kinetic energy is not a vector but a scalar. Energy as a scalar makes sense as it is only with that definition a conservative quantity:

Consider a normal pendulum where the total energy remains constant: $$ E_{tot} = E_{kin} + E_{pot} = \frac{1}{2}mv^2 + mgh = \mathrm{const}$$

Now, if one would use your definition of kinetic energy, such equation would not work anymore: for one, the potential energy is not a vector, and if it were, what direction would it be? Downward? Either way, the total energy of the system would constantly change and would be opposite when the pendulum swings left to right as compared to swinging back from right to left. While this is true for velocity or momentum, it is not useful in terms of energetic discussions.

Answer 5

No, it can't. Unlike the quantity $T=\frac{1}{2}m|\mathbf{v}|^2$, which we call kinetic energy, the quantity $\frac{1}{2}m|\mathbf{v}|\mathbf{v}$ is useless. It does not appear in any physical application nor it can be used to simplify any physics problem. You must understand that all physical quantities are named simply because they are useful, nothing more than that. If a quantity is useless, it will not have a name.

Answer 6

assume one dimensional case hence the velocity v is scalar $~v=\dot x~$

and the kinetic energy T is:

$$ T=\frac m2\,\dot x^2$$

you obtain the equation of motion

$$m\,\ddot x=-F_d$$ where $~F_d~$ is the drag force

this is also the Newton equation of motion

now your kinetic energy ansatz

$$T=\frac m2\,\dot x\,|\dot x|$$

the equation of motion is

$$ m\,\ddot x=-F_d \quad\text{if}\quad \dot x > 0$$ and $$ m\,\ddot x=+F_d \quad\text{if}\quad \dot x < 0$$

but the second equation is wrong , hence your ansatz for the kinetic energy is wrong.

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if the velocity v is a vector then $~T=\frac m2 \vec v\cdot\vec v~$ scalar

but with your ansatz $~T=\frac m2 |v|\,\vec v~$ the kinetic energy is not a scalar

Answer 7

Two no's:

1. Kinetic energy form is such $K=\frac {1}{2}mv^2$,- not because we like it to see like that, but because of relationship between elementary mechanical work and force applied $\to dW = F \cdot dr$ , the rest is just a conclusion of integration procedure, i.e. mathematical proof.

2. If you would make kinetic energy a vector in the form $\vec v \cdot |v|$, then energy conservation law would not hold anymore. Consider an ideal elastic collision of billiard ball with a steady wall. When incoming to wall - it would have kinetic energy K, and after perpendicular collision $-K$, so energy would swap for no reason. (However speed, and momentum vectors can swap). Scalar kinetic energy is very useful tool with respect to energy conservation law. If you would break that,- you would have to re-formulate 99% thermodynamics laws as well.