Question

A particle is moving in the xy-plane with velocity v(t)= vx(t)i + vy(t)j and acceleration a(t)=ax(t)i + ay(t)j. By taking the appropriate derivative show that the maginitude of v can only be constant if axvx + ayvy = 0

Answer 1

$$\frac{dv}{dt}=\frac{d}{dt}\left(\sqrt{v_x^2+v_y^2}\right)=0.5\frac{v_xa_x+v_ya_y}{\sqrt{v_x^2+v_y^2}}$$

The maginitude of v can only be constant if

$$\frac{dv}{dt}=v_xa_x+v_ya_y=0$$