Q.Derive equations of motion for a particle moving in a plane and show that the motion can be resolved into two independent motions in mutually perpendicular directions.

[!TLDR] Writing v = u + at and s = ut + (1/2)at² as vectors and resolving each into x and y components shows the x- and y-motions are governed by entirely separate, independent equations. [!ANSWER] $\vec{v}=\vec{u}+\vec{a}t$, $\vec{s}=\vec{u}t+\tfrac12\vec{a}t^2$; resolved: $v_x=u_x+a_xt$, $v_y=u_y+a_yt$, $s_x=u_xt+\tfrac12a …

Let a particle in a plane have velocity $\vec{u}$ at $t=0$ and constant acceleration $\vec{a}$. Since the acceleration is constant, average acceleration equals instantaneous acceleration, so $\vec{a}=\dfrac{\vec{v}-\vec{u}}{t}$, giving $\vec{v}=\vec{u}+\vec{a}t$ — the vector first equation of motion. For displacement, use the average velocity $\vec{v}_{av}=\dfrac{\vec{u}+\vec{v}}{2}$ (valid for constant acceleration), so $\vec{s}=\vec{v}_{av}\,t=\dfrac{\vec{u}+(\vec{u}+\vec{a}t)}{2}t=\vec{u}t+\tfrac12\vec{a}t^2$ — the vector second equation. Resolving both vector equations into x and y components gives four scalar equations: $v_x=u_x+a_xt$, $v_y=u_y+a_yt$, $s_x=u_xt+\tfrac12a_xt^2$, $s_y=u_yt+\tfrac12a_yt^2$. The x-equations involve only x-components of velocity, displacement and acceleration, and the y-equations involve only y-components — the two pairs share no common variable, so they can be solved completely separately. This proves that any motion in a plane can be treated as two simultaneous, indepe …

Generalise the rectilinear equations of motion to vector form using the definitions of average acceleration and average velocity, th …

Believing the x and y motions must somehow influence each other because they occur 'at the same time' — they are governed by completely separate equations and can be so …