# KE= 1/2 mass times velocity squared or speed squared? [closed]

I'm currently studying Physics 101 and I'm kinda lost on the subject of velocity and speed

I know that the speed is a scalar quantity which only has a magnitude, and velocity is a vector quantity which has a magnitude & direction.

but what does ($v$) refers to in $KE = 1/2 mv^2$?

I suppose its speed because we don't care about the direction in kinetic energy and in work as well; however, the internet says its velocity but I don't know why.

## closed as off-topic by AccidentalFourierTransform, M. Enns, Chris♦, Alfred Centauri, Kyle KanosFeb 15 '18 at 11:15

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• Remember that velocity is relative to some frame of reference. So for a first thing, think about what the frame is that you are measuring the KE relative to. – zeta-band Feb 14 '18 at 18:51
• @zeta-band And so is speed... – FGSUZ Feb 14 '18 at 19:00
• I'm voting to close this question as off-topic because it shows insufficient prior research. – AccidentalFourierTransform Feb 14 '18 at 19:11
• @AccidentalFourierTransform Ok, good luck – Yushi Feb 14 '18 at 19:14
• "however, the internet says its velocity" - voting to close. – Alfred Centauri Feb 15 '18 at 3:31

## 3 Answers

Both.

Let velocity be $\vec{v}$. Speed is $|\vec{v}|$. The term in KE is

$$\frac{1}{2}m\vec{v}^2=\frac{1}{2}m(\vec{v}\cdot\vec{v})=\frac{1}{2}m|\vec{v}|^2$$

• This does demonstrate a tendency which frustrated me in classes, to use "v" to describe a speed, rather than "s." As this answer shows, it's not that they used a "v" to describe a speed, so much as they got lazy drawing the || for the magnitude of the velocity vector! – Cort Ammon Feb 14 '18 at 19:21
• @CortAmmon Unfortunately for you, this tendency doesn't really go away. The notation $\vec{x}^2$ to denote a vector dotted with itself is prevalent through most of physics. – probably_someone Feb 14 '18 at 19:28
• It's not laziness. The notation $\vec v{}^2$ (or $\boldsymbol v^2$) is perfectly unambiguous and standard. At this point inserting $|\,|$ is just pedantry. Why would you use more symbols than necessary, especially when they add literally no information? – AccidentalFourierTransform Feb 14 '18 at 19:31
• @CortAmmon Well, $s$ is obviously used for "displacement" so it wasn't available for "speed"... – dmckee Feb 14 '18 at 19:55
• @Alchimista Incorrect according to whom? Landau and Lifshitz definitely use this notation, for instance. – probably_someone Feb 14 '18 at 20:38

A little about the wording of the question: as per definition, the physical quantity called „velocity” is mathematically described by a real vector space-valued function $$\vec{v} : I\subset \mathbb R\to \mathbb R^n.$$ In mathematics, vectors can be added, multiplied by scalars, but never raised to a power, be it 2,3, $\pi^e$, or any other number. Using a vector space, one can define external operations, such as an inner product $$\langle,\rangle :\mathbb R^3\times \mathbb R^3 \to \mathbb R.$$ With help of this and help of the well-defined square function in the field of real numbers, one has: $$\langle \vec{v},\vec{v}\rangle =: ||\vec{v}||^2,$$ where the quantity being squared is called norm (length) of the vector. The norm of the velocity vector is called (instantaneous) speed.

By flagrant abuse of mathematical notation, $||\vec{v}||^2$ is typically written as $\vec v^2$, or a little better as $|\vec v|^2$ which has led people to believe the velocity vector can be squared.

Bottom line, KE = 1/2 times mass of particle times (instantaneous) speed raised to the power of two.

Both are correct. In the formula $\frac{1}{2}mv^2$, direction does not matter, it is merely calculating the energy the object possess at that point in time, in whatever direction.