Direct proportionality equations

[ChiralSuperfields]

Homework Statement

Please see below

Relevant Equations

Please see below

For this,

Can a the equation $x = vt + x_0$ not be considered a direct proportionality between $x$ and $v$? If so, is it because there is a constant $x_0$?

Many thanks!

 

[hmmm27]

nope.

 

[kuruman]

Homework Statement: Please see below
Relevant Equations: Please see below

For this,
View attachment 327344
Can a the equation $x = vt + x_0$ not be considered a direct proportionality between $x$ and $v$? If so, is it because there is a constant $x_0$?

Many thanks!

The direct proportionality is between $(x-x_0)$ and $t$. If you double $v$, you double $(x-x_0)$ for the same $t$.
You can also say that there is direct proportionality between $(x-x_0)$ and $v$. If you double $t$, you double $(x-x_0)$ for the same $v$. Makes intuitive sense, no?

 

[ChiralSuperfields]

The direct proportionality is between $(x-x_0)$ and $t$. If you double $v$, you double $(x-x_0)$ for the same $t$.
You can also say that there is direct proportionality between $(x-x_0)$ and $v$. If you double $t$, you double $(x-x_0)$ for the same $v$. Makes intuitive sense, no?

Thank you for your reply @kuruman! Yeah that is interesting, and yeah I think it makes sense :)

Many thanks!

 

[Steve4Physics]

Thank you for your reply @kuruman! Yeah that is interesting, and yeah I think it makes sense :)

@kuruman's test (Post #3) for proportionality is straightforward. Ask yourself: 'If I double one quantity, does the other quantity always get doubled?'. If the answer is 'yes' the quantities are proportional.

Of course, there's nothing special about doubling. It works for any factor. E.g. if $y$ is proportional to $x$, then tripling $x$ also triples $y$. This should be clear if you thnk about the equation $y=kx$.

It’s also worth thinking graphically. If two quantities are directly proportional, a graph of one quantity against the other is a straight line through the origin.

If you get a straight line which doesn't pass through the origin, the quantities are not proportional;. In this case, the relationship is called 'linearity'. E.g. for the equation $x=vt + x_0$ there are various ways to describe the relationship between $x$ and $t$: e.g. '$x$ is linearly dependent on $t$'; or 'there is a linear releationship between $x$ and $t$'.

Edit: typo' corrected.