Putting a simple momentum equation into words

[tolove]

Check the accuracy of my wording for these first two equations, then, if you could, try to explain the last equation for me.

Ʃ F = dp/dt
"The total net force acting on a particle is equal to the rate of change of momentum over time."

∫ Ʃ F dt = ∫ dp/dt dt
"The sum of the total net force acting on a particle over a time interval is equal to the sum of the rate of change of momentum over a time interval."

My physics book breaks up differentials like this very regularly so as to make it clear for integration. But my question is, is this step a meaningful equation, or is this simply mathematical notation?
Ʃ F dt = dp

Thanks!

 

[tiny-tim]

hi tolove!

Ʃ F = dp/dt
"The total net force acting on a particle is equal to the rate of change of momentum over time."

yes, except …

"total" and "net" mean the same, so you needn't use both

personally, i'd say that "rate" means "over time", so i'd leave that out also

∫ Ʃ F dt = ∫ dp/dt dt
"The sum of the total net force acting on a particle over a time interval is equal to the sum of the rate of change of momentum over a time interval."

"sum" is definitely wrong

it's an integral (and there's no simple-english alternative to that word)

and you need to use the word "same" … "over the same time interval"

My physics book breaks up differentials like this very regularly so as to make it clear for integration. But my question is, is this step a meaningful equation, or is this simply mathematical notation?
Ʃ F dt = dp

it's mathematically meaningful (it's a statement about mathematical objects called "differentials"),

but it has no physical meaning, except as a limit or an approximation: it would never apply to an actual time interval (unless everything is constant)

 

[tolove]

hi tolove!

To correct the first one,
Ʃ F = dp/dt
"The net force acting on a particle is equal to the change of momentum over time."

and for ∫ Ʃ F dt = ∫ dp/dt dt,

"The integral of the total force acting on a particle over a certain time interval is equal to the integral of the rate of change of momentum over the same time interval."

And if I wanted to remove the word "integral," I could put the Reimann sum notation into words.

These don't make good bedtime stories. Thank you for clearing up that third equation for me! That has been driving me nuts.

 

[tiny-tim]

hi tolove!

"The net force acting on a particle is equal to the change of momentum over time."

oops!

i should have been clearer … i meant leave out "over time" … you do need the words "rate of"!

"The integral of the total force acting on a particle over a certain time interval is equal to the integral of the rate of change of momentum over the same time interval."

the integral of a rate of change is just the original function, so you could shorten that to:

"The integral of the total force acting on a particle over a certain time interval is equal to the change of momentum over the same time interval."

(and of course you can shorten "The integral of the total force" to "The impulse" )

And if I wanted to remove the word "integral," I could put the Reimann sum notation into words.

how?

(without taking several paragraphs and making it really confusing )

 

[tolove]

how?

(without taking several paragraphs and making it really confusing )

Yep! By making it long and confusing.