# B Prove mass, velocity and KE are their respected quantities

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1. Apr 2, 2016

### YES q THE zU19

I have done the example for momentum.

And I gather that scalar*vector=vector.

I know that mass and KE is scalar, velocity is vector.

Can someone show me proofs like for what I have said above.

Not just mass is scalar because it does not have direction etc.

Thank you.

2. Apr 2, 2016

### Staff: Mentor

KE is defined as $KE=1/2 m v^2$, where $m$ is a scalar and $v$ is a vector. $v^2$ is short for $v \cdot v$ which is the dot product, an operation which takes two vectors and returns a scalar. So although $v$ is a vector $v^2$ is a scalar, and thus KE is a scalar.

3. Apr 3, 2016

### YES q THE zU19

Thank you, this was what I was looking for.

So in general we have;

scalar*scalar= scalar?

scalar*vector = vector

vector*vector = scalar.

Work done = energy

So force * distance = vector * scalar? = vector

Could you also kindly tell me about how to prove mass is scalar please.

4. Apr 3, 2016

### jbriggs444

Work is force times displacement. Displacement is a vector, not a scalar. The product is a dot product and produces a scalar.

Mass is defined as a scalar. It is a scalar by definition.

5. Apr 3, 2016

### Staff: Mentor

Not always. There are two vector products. The dot product takes two vectors and gives a scalar, but the cross product takes two vectors and gives another vector. These are usually written as $a \cdot b$ and $a \times b$ respectively.

For the rest of your questions I agree with jbriggs444's answers above, particularly for Newtonian mechanics.

6. Apr 3, 2016

### DrStupid

And the tensor product results in a matrix.

7. Apr 4, 2016

### Staff: Mentor

Don't forget a useful property of basic arithmetic. For real numbers, even powers are always positive, odd powers can be plus or minus. Vectors, like velocity, need to have direction and thus change sign. Scalars, like temperature or speed, have no sign.

That is not physics, but it can be useful in physics. For example, $mv^2$ is always positive. It takes the same energy to accelerate a body to an eastward velocity as to a westward velocity. You can spot that instantly because the power 2 is even.

8. Apr 12, 2016

### YES q THE zU19

Thank you.

9. Apr 12, 2016

### YES q THE zU19

Thanks dale.

10. Apr 12, 2016

### YES q THE zU19

Thank you for this.