Question on Feynman's Proof of Energy Conservation

If $v$ is treated as the velocity instead of the speed of the object then $m(dv/dt)$ has direction and would be simply the rate of change of momentum, rather than the rate of change of the magnitude of the momentum.

The reason for this imprecision probably stems from the two definitions of kinetic energy: it can be calculated as $frac{1}{2}mv^2$, where $v$ is the speed of the object, or $frac{1}{2}m(textbf vcdottextbf v)$ where $textbf v$ is the velocity of the object.

If we use the first definition then the rate of change of kinetic energy is dependent on $m(dv/dt)$, which is directionless.

If we use the second definition then it is dependent on $m(dtextbf v/dt)$ which has direction, and is therefore not the rate of change of the magnitude of the momentum, but simply the rate of change of the momentum.

Feynman does not use vector analysis till later in the chapter, so he is precise in saying that $m(dv/dt)$ is the rate of change of the magnitude of the momentum.

But shouldn't he then write the "magnitude of the force in the direction of motion", instead of the "force in the direction of motion"? No. Since a component of a vector in a particular direction is a scalar, so it is superfluous to say the magnitude of a scalar.

Hopefully this helps.