I never understood the derivation of centripetal acceleration...until now!

Derivation of Centripetal Acceleration Formula
📌 The video provides an intuitive explanation for why centripetal acceleration ($a_c$) is proportional to $\frac{v^2}{r}$.
🔄 Centripetal acceleration is defined as the change in velocity ($\Delta v$) divided by the time taken ($\Delta t$): $a_c = \frac{\Delta v}{\Delta t}$.
🔺 When the car's speed ($v$) doubles, the velocity vectors also double, causing the change in velocity ($\Delta v$) to double. Simultaneously, the time taken ($\Delta t$) to travel the same angle is halved, resulting in acceleration increasing by a factor of $2 / (\frac{1}{2}) = 4$, or $2^2$.

Impact of Speed on Acceleration
🏎️ Doubling the speed results in $\Delta v$ doubling and $\Delta t$ becoming one-half, leading to an acceleration increase of four times ($2^2$).
⏫ Tripling the speed results in $\Delta v$ tripling and $\Delta t$ becoming one-third, leading to an acceleration increase of nine times ($3^2$).
📈 The acceleration is shown to be directly proportional to the square of the speed ($v^2$) due to the combined effect of changes in both $\Delta v$ and $\Delta t$.

Impact of Radius on Acceleration
⭕ Doubling the radius ($r$) while keeping speed constant causes $\Delta v$ to remain exactly the same because the angle of rotation is unchanged.
⏳ Doubling the radius means the distance traveled (arc length $AB$) doubles, so the time taken ($\Delta t$) to cover this distance at the same speed also doubles.
📉 With the same $\Delta v$ occurring over twice the time, the acceleration becomes one-half ($\frac{1}{2}$), demonstrating that acceleration is inversely proportional to the radius ($a_c \propto \frac{1}{r}$).

Key Points & Insights
➡️ Centripetal acceleration ($a_c$) depends on velocity ($v$) squared and radius ($r$) inversely, leading to the relationship $a_c \propto \frac{v^2}{r}$.
➡️ The $v^2$ term arises because increasing velocity simultaneously increases the change in velocity ($\Delta v$) and decreases the time interval ($\Delta t$) proportionally.
➡️ Changes in the radius ($r$) affect only the time required to traverse the arc, as $\Delta v$ depends only on the angular change, not the radius magnitude.

📸 Video summarized with SummaryTube.com on Feb 13, 2026, 07:36 UTC