Displacement and distance when particle is moving in curved trajectory

[haruspex]

Thanks for your reply @haruspex , I guess the reason why there are no rules for adding a vector to a scalar is because it is not physically meaningful yet. Maybe in the future of physics!

It is a matter of mathematics, not physics.

 

[member 731016]

It is a matter of mathematics, not physics.

Ok thanks for your reply @haruspex !

 

[PeroK]

Thanks for your reply @PeroK , what do you mean same 'order'. Many thanks!

Scalar, vector or tensor generally.

 

[member 731016]

Scalar, vector or tensor generally.

Ok thank you @PeroK!

 

[kuruman]

To me the "$=$" sign is shorthand notation for "is the same as" so that I don't have to write out the entire locution in English every time I put an equation down.

In mathematics, we write the vector equation $\mathbf{A}=\mathbf{B}$. Bereft of symbolic notation, this says, in English, "Vector A is the same as vector B". If we want to qualify this some more, we say "This means that the x-component of vector A is the same as the x-component of vector B and the y-component of vector A is the same as the y-component of vector B and the z-component of vector A is the same as the z-component of vector B." All that verbosity can be eliminated by writing "$A_x=B_x;~A_y=B_y;~A_z=B_z.$"

In physics, which uses mathematics as its language, the "$=$' is still shorthand for "is the same as" but more nuanced because it also provides a correspondence between observables. For example, I can use a scale to measure mass $m$. I can then apply to it a constant force $F$, which I can measure with a force gauge, and figure out the mass's constant acceleration $a$ by measuring its positions with a ruler and the time at these positions with a clock. Then I can write down the number from the force gauge and, next to it, the scale reading for the mass with the result of my calculation for the acceleration. Lo and behold, the product of the mass and the acceleration is the same as (to within measurement uncertainty) the reading of the force gauge.

In this context, "Net force equals mass times acceleration" says less than "Net force is the same as mass multiplied by acceleration." The former implies that if I give you numbers of the mass and the acceleration, you can find a number for the net force. The latter implies that if you measure all quantities that participate in the equation on both sides separately and then put them together as prescribed, the left-hand side will be the same as the right hand side. We think of the former as a definition and the latter as a "law". One tends to forget that equations in physics labeled "laws" are based on painstaking measurements.

 

[member 731016]

To me the "$=$" sign is shorthand notation for "is the same as" so that I don't have to write out the entire locution in English every time I put an equation down.

In mathematics, we write the vector equation $\mathbf{A}=\mathbf{B}$. Bereft of symbolic notation, this says, in English, "Vector A is the same as vector B". If we want to qualify this some more, we say "This means that the x-component of vector A is the same as the x-component of vector B and the y-component of vector A is the same as the y-component of vector B and the z-component of vector A is the same as the z-component of vector B." All that verbosity can be eliminated by writing "$A_x=B_x;~A_y=B_y;~A_z=B_z.$"

In physics, which uses mathematics as its language, the "$=$' is still shorthand for "is the same as" but more nuanced because it also provides a correspondence between observables. For example, I can use a scale to measure mass $m$. I can then apply to it a constant force $F$, which I can measure with a force gauge, and figure out the mass's constant acceleration $a$ by measuring its positions with a ruler and the time at these positions with a clock. Then I can write down the number from the force gauge and, next to it, the scale reading for the mass with the result of my calculation for the acceleration. Lo and behold, the product of the mass and the acceleration is the same as (to within measurement uncertainty) the reading of the force gauge.

In this context, "Net force equals mass times acceleration" says less than "Net force is the same as mass multiplied by acceleration." The former implies that if I give you numbers of the mass and the acceleration, you can find a number for the net force. The latter implies that if you measure all quantities that participate in the equation on both sides separately and then put them together as prescribed, the left-hand side will be the same as the right hand side. We think of the former as a definition and the latter as a "law". One tends to forget that equations in physics labeled "laws" are based on painstaking measurements.

Thank you very much for your answer @kuruman !

 

[member 731016]

In this context, "Net force equals mass times acceleration" says less than "Net force is the same as mass multiplied by acceleration."

@kuruman Could you please clarify more how "Net force is the same as mass multiplied by acceleration" says more than "Net force equals mass time acceleration"?

Many thanks!