Motion of an object that always has an acceleration perpendicular to its velocity

Question

Consider an object whose position vector $x$(changes with time) satisfies the condition $$ \dot{x}\cdot\ddot{x}=0 $$ i.e. the object is always accelerating in the direction perpendicular to its direction of motion.

How can I solve the DE and find an equation relating $x$ and $t$(the time)? Is it wrong to say that the speed should be constant?

Answer 1

For simplicity, consider the problem to be two dimensional.

Let $\mathbf v=v_x \mathbf i+v_y \mathbf j$.

So you want to solve $$\mathbf v\cdot\mathbf {\dot v}=0$$

This is equivalent to solving $$v_x \dot v_x+v_y \dot v_y=0\qquad{(1)}$$

However, in general, $v_x$ and $v_y$ are independent of each other. With one equation and two unknowns, our problem is underdetermined.

However, it can be mathematically proved that the speed remains constant.

Integrating $(1)$ with respect to $t$ yields $$v_x^2+v_y^2=C\implies ||\mathbf v||=\text{constant}$$

Answer 2

Yes the speed must be constant. This is just the case of circular motion and yes the speed is constant for a circular motion.