# Why tangential acceleration become 0 when the velocity is max?

I Know that tangential acceleration equal to zero when the circular motion is uniform, but why it is equal to 0 , when the velocity is max or min , because there is no relation between the value of velocity and derivation since the velocity change, and if the velocity is max is that mean that the motion become uniform?

## 2 Answers

You need to be a little careful in questions like these.

when the velocity is max or min

This is not a meaningful statement. Velocity is a vector and there is no meaningful max or min for a vector quantity.

What you are really interested in is the speed, which is the magnitude of the velocity, not the velocity. You can have a max or min speed, but not a max or min velocity. Particularly in circular motion where the direction is constantly changing.

So, the actual question you want answered is

why is the tangential acceleration equal to zero when the speed is max or min?

The speed is the magnitude of the velocity $$v=\sqrt{\vec v \cdot \vec v}$$ so it has a maximum or a minimum when $$\frac{dv}{dt}=\frac{d}{dt}\sqrt{\vec v \cdot \vec v} = \frac{\vec a \cdot \vec v}{v}=0$$ Now, in order to even be able to define a tangential direction we require that $$v\ne 0$$. So this means that for the speed to be a min or max requires that $$\vec a \cdot \vec v =0$$

Now, the tangential component of the acceleration is the component of $$\vec a$$ that is in the direction parallel to $$\vec v$$. That can be written as $$\vec a_{\parallel} = (\vec a \cdot \vec v) \hat v$$ and since $$\vec a \cdot \vec v = 0$$ then $$\vec a_{\parallel}=0$$ also.

Consider for intance a central motion, where the force, and thus the acceleration, is directed toward a fixed center. The absolute value of the velocity reaches its maximum or minimum value when $$z(t):= \vec{v}(t)\cdot \vec{v}(t)$$ does. In that instant, the function $$z$$ has zero detivative. Computing the derivative: $$\frac{dz}{dt} = 2\vec{v} \cdot \vec{a} =0$$ where the acceleration is $$\vec{a}$$. You see that the velocity is normal to the acceleration in that instant. It has no component along the velocity: the tangential acceleration is zero.