In order to get a good sense of acceleration at a specific instance in time, we

Question

In order to get a good sense of acceleration at a specific instance in time, we would have to look at the velocity at that point in time and at a time very soon thereafter (read, on the order of tenths or hundredths of a second). For example, to get an idea of the acceleration at 0.5 seconds into the run, we would need velocity at 0.5 seconds and also at 0.51 seconds (assuming we can measure velocity on this time scale). We would then need to perform this analysis for each point in time. We could use then use all the acceleration calculuations to generate a graph for acceleration at each point in time. Statistics would then allow us to determine the best math function to use to describe the acceleration of the car at each point in time. It's a little tedious, but it would be one way to get a math equation for acceleration.

I would curious to know more about the assumption that the acceleration of the car is linear. Do you have any ideas why this assumption was made? If not, it's okay, as this is more of an engineering/phyics questions, but I am certainly curious to know more!

Explanation / Answer

This is easy enough to explain.

Take any random function say y= x^2
There are two points (1,1) and (1.1 , 1.21)

Now, draw a line between these points....

Does it not look the parabola is traversing a line between these extremely close points?

Yes, it does.

Now, in reality when we are finding the acceleration, we find
the v at t = 0.5 and t = 0.51, which is insanely close to each other, therefore, it is ok to assume that it is a line and simply find the slope to find the instantaneous acceleration at any particular instant.

Hope this helped!

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