# I What is instantaneous acceleration?

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1. May 8, 2017

### parshyaa

How their can be instantaneous acceleration, its impossible to have change in velocity at a particular position(instant), we can have velcoity or speed at a particular point but how can we have change in velocity at a particular instant?

2. May 8, 2017

### andrewkirk

We don't.

Acceleration is simply the derivative of velocity with respect to time. Spend some time pondering the definition of derivative as a limit as $\delta t\to 0$, and how it applies to this particular case, and you'll see how it works.

3. May 8, 2017

### parshyaa

You mean Δt→0

4. May 8, 2017

### Ssnow

Hi, from your velocity function $v(t)$ respect the time $t$ you can define the instantaneous acceleration in $t=t_{0}$ by the derivative:

$a(t_{0})=\lim_{\Delta t \rightarrow 0}\frac{v(t_{0}+\Delta t)-v(t_{0})}{\Delta t}$

Ssnow

5. May 8, 2017

No I don't.

6. May 8, 2017

### parshyaa

@Ssnow

Last edited: May 8, 2017
7. May 8, 2017

### parshyaa

I think we defined instantaneous acceleration mainly to use its vector property so that we can determine in which direction velcoity is increasing and its magnitude will be the change in velocity at a very small interval(and we say it as change in velocity at an instant or acceleration at an instant), therefore we never use scalar version or we didn't defined scalar version of acceleration.
Its just my attempt to answer this question.

8. May 8, 2017

### jbriggs444

If we can define instantaneous velocity as the instantaneous rate of change of position with respect to time then how is it problematic to define instantaneous acceleration as the instantaneous rate of change of velocity with respect to time?

9. May 8, 2017

### Ben Wilson

when you first get in your car, a=0. You turn the key, a=0. You start to move, a=10 (pick any unit you want) then a few seconds later now the wheels are going faster a=40, then the wheels can't really go any faster so a=20, and at top speed when you can't get any faster a=0. It's useful to measure acceleration as an average, e.g. from start to a top speed of 60mph, we might say the car went from 0 to 60 in so many seconds. But when breaking down the motion into separate instances, we have different accelerations. This is the concept of instantaneous acceleration, i.e. the acceleration at this or that moment, as opposed to an average acceleration.

10. May 8, 2017

### parshyaa

So the final definition becomes,

1. instantaneous velcity is the velocity of the particle at an instant{its completely logical because we can find the velcity of a particle at a particular instant(time)}
2. Instantaneous acceleration is the rate of change of velcity of a particle at a particular instant(this is not logical to me, how can we have change in velcoity at a particular instant)
As you said that whats the problem in defining this way, i didn't said that we can't define acceleration as rate of change of velocity(i think this definition is good for average acceleration but not for instantaneous acceleration as for above reasons)

11. May 8, 2017

### Ben Wilson

thats the simple physics definition. If you then have an issue with the math... v=dx/dt, a=dv/dt, they are the same mathematical objects: a derivative w.r.t a variable t; where x, v, and a are the vector quantities displacement, velocity and acc respectively.

12. May 8, 2017

### Ben Wilson

If you replace in 2. "rate of change of velocity of a particle" with "acceleration" .OR. replace in 1. "velocity of" with "rate of change of displacement of". Do you still have the same conceptual problem?

13. May 8, 2017

### A.T.

Note how the red part is missing in your question.

Seems like your problem is with math, not with physics:
https://en.wikipedia.org/wiki/Differential_calculus

And your logic makes no sense to me. Acceleration is a derivative of velocity, just like velocity is a derivative of position. Both have a instantaneous value at every time point.

14. May 8, 2017

### parshyaa

How can we have a change in velcity at an instant(or acceleration at an instant)

15. May 8, 2017

### A.T.

See post #13 again, I edited it.

16. May 8, 2017

### parshyaa

Do you mean that accelaration is the velocity at a instant of time and velocity is the position of a particle at a instant of time.

17. May 8, 2017

### Ben Wilson

same way you can have change in position at an instant.
no. you need to learn calculus, which will define what an instantaneous rate of change is.

18. May 8, 2017

### jbriggs444

How, experimentally, will you find the velocity of a particle at a particular instant?

Mathematically, we can define it -- no problem.

19. May 8, 2017

### parshyaa

By drawing a tangent at that instant

20. May 8, 2017

### Ben Wilson

impossible

21. May 8, 2017

### parshyaa

Okk i got it, we can't find velocity at a particular instant experimentally therefore we use calculus, whats the point.

22. May 8, 2017

### Ben Wilson

that dv/dt and dx/dt are the same things mathematically, so what is your problem with them behaving the same?

23. May 8, 2017

### parshyaa

Actually my definition of accelaration was wrong, i always thaught that accelaration is the change in velocity w.r.t time,but correct definition is the ability to gain velocity is called acceleration, and therefore we can have acceleration at an instant, sorry for creating mess. Thank you so much

24. May 8, 2017

### Ben Wilson

that is true... a=dv/dt

25. May 8, 2017

### parshyaa

Yep both definitions are corrrct but the 2nd definition which i found on google explains completely that we can have accelaration at an instant.Accelaration is the change in velcoity w.r.t time but here change must be positive or velcity must increase. Therefore we can define instantaneous acceleration as the ability to increase its speed or gain its speed at an instant