Recently I asked this question that how perpendicular acceleration can change the direction of velocity without changing its magnitude, one of the explanation was for a small interval of time that perpendicular acceleration will change the velocity of the object in the direction of acceleration that is given by this equation $$V_i + a \ dt = V_f$$ (where $V_i$ and $V_f$ are the initial and final velocity of the particle and $a$ is the perpendicular acceleration) as the gain in velocity(magnitude) is very small and the force acts on the particle in that particular direction for a very small time dt and also it is a differential quantity it can be ignored my doubt is:-
1)The change in direction is also very small why cant we ignore it means how can adt change the direction of velocity without changing magnitude especially if acceleration is perpendicular to velocity.
2)That increase in magnitude is a small quantity but it is getting added at every instance so its integration is non zero. So why are we ignoring it.
Best Answer
$$|\mathbf v|^2=\mathbf v\cdot \mathbf v$$ $$\frac{\text d}{\text dt}\left(|\mathbf v|^2\right)=2\mathbf v\cdot\frac{\text d\mathbf v}{\text dt}$$
As you can see, if the acceleration $\mathbf a=\text d\mathbf v/\text dt=0$, then the magnitude of the velocity vector $\mathbf v$ will not change.
You're constraining the acceleration to always be perpendicular to the velocity. So the "small quantity" is in a different direction at each instant, as the velocity direction changes at each instant. You are right that if you only had the acceleration perpendicular to the velocity at only one instant and then kept the acceleration fixed there, then the velocity would also change magnitude at future times. (For example, think of the velocity of a projectile at the top of its trajectory and then as it starts to fall back down).