Prove mass, velocity and KE are their respected quantities

[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.

 

[Dale]

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.

 

[YES q THE zU19]

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.

Thank you, this was what I was looking for.

So in general we have;

scalar*scalar= scalar?

scalar*vector = vector

vector*vector = scalar.

What about work done though?

Work done = energy

So force * distance = vector * scalar? = vector

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

 

[jbriggs444]

Work done = energy
So force * distance = vector * scalar? = vector

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

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

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

 

[Dale]

vector*vector = scalar.

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.

 

[DrStupid]

The dot product takes two vectors and gives a scalar, but the cross product takes two vectors and gives another vector.

And the tensor product results in a matrix.

 

[anorlunda]

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.

 

[YES q THE zU19]

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.

Thank you.

 

[YES q THE zU19]

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.

Thanks dale.

 

[YES q THE zU19]

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.

Thank you for this.