Equations of Motion for an Object Travelling in a Plane with Uniform Acceleration

Section 3.2 derived the equations of motion for a rectilinear object moving with uniform acceleration; this section derives the corresponding equations for a particle moving in a plane with a constant (uniform) acceleration, by simply writing the earlier one-dimensional derivation in vector form.

Let the particle's velocity be $\vec{u}$ at $t=0$ and $\vec{v}$ at time $t$. Since the acceleration $\vec{a}$ is constant, the average and instantaneous accelerations coincide, so by the definition of average acceleration, $\vec{a} = \dfrac{\vec{v}-\vec{u}}{t-0}$, which rearranges to the first equation of motion in vector form: $\vec{v} = \vec{u} + \vec{a}t$ — the vector analogue of Eq. (3.7).

For the displacement $\vec{s}$ from $t=0$ to $t$, use the average velocity: for constant acceleration $\vec{v}_{av} = \dfrac{\vec{u}+\vec{v}}{2}$, so $\vec{s} = \vec{v}_{av}\,t = \dfrac{\vec{u}+\vec{v}}{2}t = \dfrac{\vec{u} + (\vec{u}+\vec{a}t)}{2}t$, which simplifies to the second equation of motion in vector form: $\vec{s} = \vec{u}t + \tfrac{1}{2}\vec{a}t^2$ — the vector analogue of Eq. (3.8). …