AHL 3.12—Vector applications to kinematics

Kinematics is the branch of mechanics that describes how objects move , without worrying about the forces causing that motion. In AHL 3.12, we apply vector methods to kinematics in two and three dimensions, giving us a unified, elegant framework for analysing all kinds of motion.

The three key vector quantities you need to know are:

• Position vector $\mathbf{r}(t)$: the location of an object at time $t$
• Velocity vector $\mathbf{v}(t)$: the rate of change of position, $\mathbf{v}(t)=\frac{d\mathbf{r}}{dt}$
• Acceleration vector $\mathbf{a}(t)$: the rate of change of velocity, $\mathbf{a}(t)=\frac{d\mathbf{v}}{dt}$

Throughout this subtopic, we work with both constant velocity (linear motion) and variable velocity (e.g. projectile motion, circular motion), connecting differentiation and integration to physical motion.

When an object moves with constant velocity, its position changes linearly with time. The vector equation of motion is:

$\mathbf{r}(t)={\mathbf{r}}_0+\mathbf{v}t$

where:

• $\mathbf{r}(t)$ is the position vector at time $t$
• ${\mathbf{r}}_0$ is the initial position vector (position at $t=0$)
• $\mathbf{v}$ is the constant velocity vector
• $t$ is time (usually in seconds)

In three dimensions, this expands to:

$\mathbf{r}(t)=\left(\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right)+\left(\begin{array}{c}{v}_x\\ v_y\\ v_z\end{array}\right)t$

Each component is independent: $x(t)=x_0+v_{x}t$, $y(t)=y_0+v_{y}t$, $z(t)=z_0+v_{z}t$.

Velocity Vector: The velocity vector $\mathbf{v}$ gives both the speed (its magnitude $|\mathbf{v}|$) and direction of motion. For constant velocity motion, $\mathbf{v}$ does not change with time.