Does Acceleration Depend on Instantaneous Velocity Changes?

[Carbon273]

This question is not really a problem but a conversation starter. I am studying physics for the fun of it and I do have a quick question. Does acceleration regardless if it is average or instantaneous depend upon the change in the instantaneous velocity? The reason I asked that is because most books that I have seen does not specify the velocity to being instantaneous (hence why I underlined it). Most books just say the change of velocity over time is the cause with no specification. I know this is nitpicky and an obvious question but it has been bothering me. Thanks for the help in advance.

 

[kuruman]

Hi Carbon273 and welcome to PF.

Does acceleration regardless if it is average or instantaneous depend upon the change in the instantaneous velocity?

Yes. Average acceleration is the change in velocity (final minus initial instantaneous velocity) divided by the time interval over which this change takes place. Instantaneous acceleration is the same ratio but in the limit that the time interval becomes infinitesimally small, i.e. it is the slope of the velocity vs. time graph at some specific time.

 

[Bystander]

https://thespectrumofriemannium.wordpress.com/2012/11/10/log053-derivatives-of-position/

 

[Carbon273]

Hi Carbon273 and welcome to PF.Yes. Average acceleration is the change in velocity (final minus initial instantaneous velocity) divided by the time interval over which this change takes place. Instantaneous acceleration is the same ratio but in the limit that the time interval becomes infinitesimally small, i.e. it is the slope of the velocity vs. time graph at some specific time.

That's interesting. Obviously, its an assumption the books wants you to have but clarification would help on their part. Thanks for the help.

 

[sophiecentaur]

Instantaneous acceleration is the same ratio but in the limit that the time interval becomes infinitesimally small

This is the basis of differential calculus. It works for any relationship between variables where the function is "continuous and differentiable" within the limits you are considering. This actually means most basic ideas encountered in Physics except where the relationship 'goes to infinity' (very bad maths statement but it gives the message) or when there is a 'cusp' in the curve.
Whenever we do a set of measurements or spot calculations, we sort of assume that it's valid to draw a smooth curve through them and it's mostly ok for many Physics experiments. The magic word here is "LIMIT".
PS Phenomena involving Chaotic Behaviour cannot be treated this way because there is no continuous line relating Input and Output result.