[Physics] What’s the difference between average velocity and instantaneous velocity

calculusdifferentiationkinematicsvelocity

Suppose the distance $x$ varies with time as:
$$x = 490t^2.$$
We have to calculate the velocity at $t = 10\ \mathrm s$.
My question is that why can't we just put $t = 10$ in the equation $$x = 490t^2$$ which gives us total distance covered by the body and then divide it by 10 (since $t = 10\ \mathrm s$) which will give us the velocity, like this:-
$$v~=~\frac{490 \times 10 \times 10}{10} ~=~ 4900\ \frac{\mathrm{m}}{\mathrm{s}}$$
Why we should use differentiation, like this:
$$
\begin{array}{rl}
x & = 490t^2 \\
\\
v & = \mathrm dx/\mathrm dt \\
& = \mathrm d(490t^2)/\mathrm dt \\
& = 490 \times 2 \times t \\
& = 490 \times 2 \times 10 \\
& = 9800\, \frac{\mathrm{m}}{\mathrm{s}}
\end{array}
$$
Which not only creates confusion but also gives different answer.
Any help is highly appreciated.

Best Answer

Your question is legitimate and I don't understand why it got downvoted. The confusion arises in the difference between average and instantaneous velocity.

Consider this example: a car moves at 10 m/s for 5 seconds, then stops at a light for another five seconds. What is the velocity of the car after 7 seconds? According to your calculation, it would be $\frac{5 \,\textrm{s}\cdot10\,\textrm{m/s}}{7 \, \textrm{s}}\approx 7.14$ m/s, which is obviously wrong because the car is completely at rest after 7 seconds. What you just computed is the average velocity of the car during those 7 seconds.

Asking for the velocity of a body at a given point in time is equivalent to asking "how much will the position change after an infinitesimal amount of time?", which is, in non rigorous terms, like taking an infinitesimal amount of space $dx$ and dividing it by an infinitesimal amount of time $dt$ (this is not how derivatives actually mathematically are defined, but it works at an intuitive level). The average velocity during an infinitesimal amount of time becomes the instantaneous velocity and is computed using the derivative.

in our previous example we would obtain $0$, because at 7 seconds, and just before and just after 7 seconds, the car is at rest.

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