The H2 syllabus does not require students to derive . But the derivation was an eye-opener to me and I still remember the sense of wonder when I first read it when I was 17 years old.

Let’s consider an object travelling at constant speed *v* along a circular arc of radius *r*. The object travels from *s*_{1} to *s*_{2} in time duration , undergoing a displacement of and an angular displacement of .

Since we are geniuses, we detect two isosceles triangles both subtended by angle Δ*θ*. The first one, which I call the displacement triangle, is formed by the displacement vectors *s*_{1} and *s*_{2}. It has two equal sides *r*, and the third side represents Δ*s*. The second triangle, which I call the velocity triangle, is formed by the velocity vectors *v*_{1} and *v*_{2}. It has two equal side *v*, and the third side represents the change in velocity Δ*v*. (Think vector subtraction)

Since these two triangles are similar triangles, we can write down

Plot twist. Since we are looking for the instantaneous acceleration, we should be looking at an infinitesimally small Δ*t*. Meaning an infinitesimally small Δ*θ*. Meaning our isosceles triangles should actually be needle-shaped wedges, with Δ*θ* approaching 0°, and the other two angles approaching 90°.

Let’s examine the velocity triangle first. Do you realize that as Δ*θ* approaches 0°, Δ*v* becomes perpendicular to velocity? Since acceleration is in the direction of velocity change, this confirms that the acceleration (during uniform circular motion) is in the centripetal direction.

We now examine the displacement triangle. Do you realize that as Δ*θ* approaches zero, the circular arc *v*Δ*t* becomes indistinguishable from the chord Δ*s*?

Going back to earlier equation, and substituting Δ*s* for *v*Δ*t*, we have

Isn’t it awesome? To look for the instantaneous , we look for when , which inevitably means that . So is a near zero number, divided by another near zero number. The outcome is neither zero nor infinity. It is . Welcome to circular motion.

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**Video Explanation**

The Similar Triangle Proof for *v*^{2}/*r*

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