Understanding Products in the Equation v^2 = u^2 + 2aS

[johncena]

In the equation v^2 = u^2 + 2aS , What kind of products are v^2 , u^2 , and aS ?
Cross product or dot product?

 

[Ask1122]

Please correct me if i am wrong, but i believe they will be dot products, cross products will have a change in directions as well.

 

[Fightfish]

Definitely dot products.
When you cross a vector with itself, you get the zero vector, which is absolutely meaningless.

 

[johncena]

OK . So v^2 and u^2 are dot products ...but what about aS?

 

[Fightfish]

OK . So v^2 and u^2 are dot products ...but what about aS?

Going from just a shallow point of view (without analyzing the meaning of the equation whatsoever) it must be a dot product as well as v^2 and u^2 are both scalars, which necessarily requires the product aS to yield a scalar as well.

 

[D H]

Unless of course a and S are already scalars.

 

[Gerenuk]

I think it is important to not just guess what the products might be, but rather prove the law anew. It might be none of the products. So let's do that
\Delta E_\text{kin}=\int\vec{F}\cdot\mathrm{d}\vec{s}
\therefore m|v|^2-m|u|^2=2\vec{F}\cdot\Delta\vec{s}
if the force is a constant
\therefore |v|^2=|u|^2+2\vec{a}\cdot\Delta\vec{s}
or if you wish
\therefore \vec{v}\cdot\vec{v}=\vec{v}_0\cdot\vec{v}_0+2\vec{a}\cdot(\vec{s}-\vec{s}_0)

Note that all this assumes that the force/acceleration is constant.

 

[Redbelly98]

Note that all this assumes that the force/acceleration is constant.

[nitpick]

Just to clarify, Gerenuk means that force and acceleration are both constant.

\frac{force}{acceleration}[/itex] is the same as the mass, which is always constant (at nonrelativistic speeds)<br /> <br /> <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":smile:" title="Smile :smile:" data-smilie="1"data-shortname=":smile:" /><br /> [/nitpick]