regarding instantaneous acceleration

Question

instantaneous acceleration means v1(velocity at t1) and v2(velocity at t2) happen at same instant, i.e v1 and v2 are considered to happen at same time i.e t1=t2. am i right?

Answer 1

Kind of. Acceleration is the derivative of velocity and mathematically, it means the acceleration (instantaneous acceleration) at some time t₁ is the limit as h approaches zero of [v(t₁ + h) - v(t₁)]/[(t₁+h) - t₁]. So, in your nomenclature t₂ = t₁ + h in my equation. And you are thinking about the slope of the secant line between two points (t₁, v(t₁)) and ((t₁ + h), v(t₁ + h)) on the velocity curve as the points get closer and closer together. BUT, in calculus you never actually get to the point where t₁ and t₂ are the same, you just keep getting infinitely closer.

So it's perhaps better to think about t₁ and t₂ being separated by an infinitesimally small amount of time.