The speed of a particle moving in a plane is equal to the magnitude of its…

PHYSICS · Multi-part · AP · Physics C: Mechanics · Kinematics · Kinematics

Problem

The speed of a particle moving in a plane is equal to the magnitude of its instantaneous velocity:

v=|\mathbf{v}|=\sqrt{v_x^2+v_y^2}.

(a) Show that the rate of change of the speed is

\frac{dv}{dt}=\frac{v_xa_x+v_ya_y}{\sqrt{v_x^2+v_y^2}}

. (b) Show that the rate of change of speed can be expressed as

\frac{dv}{dt}=\frac{\mathbf{v}\cdot\mathbf{a}}{v}

, and use this result to explain why

\frac{dv}{dt}

is equal to

a_t

, the component of

\mathbf{a}

that is parallel to

\mathbf{v}