It comes from the fact that velocity is a vector and you can change your velocity vector by changing your direction.

其原理就是,速度是矢量,你可以通过改变方向来改变速度

耶鲁公开课 - 基础物理课程节选

That limit will be some arrow we can call the velocity at the time and it will always be tangent to the curve.

那个极限也就是一个矢量,我们称之为瞬时速度,并且它总是和轨迹相切的

耶鲁公开课 - 基础物理课程节选

The tangent's pointing towards the direction you are headed at that instant.

切线的方向就是,速度矢量在那个瞬时的指向

耶鲁公开课 - 基础物理课程节选

Just take the derivatives and that's the velocity vector.

直接计算对时间的导数就是速度矢量

耶鲁公开课 - 基础物理课程节选

you get the velocity vector.

就可以得到速度矢量

耶鲁公开课 - 基础物理课程节选

The fact that when you go in a circle, you accelerate is what we're learning here, coming from the fact that velocity is a vector and its change can be due to change in the magnitude or change in direction.

而当你做圆周运动时你也在加速,这是我们在这里所学到的,原因就在于,速度是一个矢量,其变化可以通过改变模长,或者方向来实现

耶鲁公开课 - 基础物理课程节选

take the derivative of this, get the velocity vector and you notice his magnitude is a constant Whichever way you do it, you can then rewrite this as v square over r.

对这个式子求一次导,就能得到速度矢量,你会发现其模长是常数,不管用什么方法,加速度也可以写成 v^2 / r

耶鲁公开课 - 基础物理课程节选