A-Level Maths: Vectors in Kinematics

In the second year of A level Mathematics, kinematics moves from a straight line into a plane. Position, velocity and acceleration become vectors written with $\mathbf{i}$ and $\mathbf{j}$, and the calculus you already know is applied to each component. This material trains that extension, based on the DfE A level Mathematics subject content on kinematics (using calculus for motion with variable acceleration, and extending to two dimensions using vectors) together with Newton's second law in vector form.

You differentiate a position vector $\mathbf{r}(t)$ to find the velocity and acceleration, and integrate an acceleration or velocity vector back again, using a known velocity or position to fix the vector constant of integration. You find speed as the magnitude of the velocity, describe a direction of motion as a bearing when $\mathbf{i}$ and $\mathbf{j}$ point east and north, and interpret conditions such as moving parallel to $\mathbf{i}$ (the $\mathbf{j}$ component of velocity is zero) or being instantaneously at rest (every component is zero at the same time).

For constant acceleration you use the vector forms $\mathbf{v}=\mathbf{u}+\mathbf{a}t$ and $\mathbf{r}={\mathbf{r}}_0+\mathbf{u}t+\frac{1}{2}\mathbf{a}{t}^2$, and you connect motion with forces through $\mathbf{F}=m\mathbf{a}$. Other problems ask whether two moving particles meet, which needs every component to agree at the same time, and model a projectile with a single position vector, reading off when it lands and how far it travels.

Each question gives all the data you need, with $g=9.8{\text{ m s}}^{-2}$ where gravity is involved.

The material offers three formats. The quiz has 12 worked multiple-choice problems whose explanations show each differentiation or integration step and the typical mistakes behind the wrong answers, such as forgetting the initial position or measuring a bearing from the wrong axis. The flashcards summarise the key relationships and conditions. The written work gives 8 longer problems to solve by hand, including integrating twice from an acceleration, two ships moving on the same map and a projectile launched above the ground.

• Differentiate a position vector to find velocity and acceleration
• Integrate acceleration and velocity vectors using initial conditions
• Find speed and direction of motion, including bearings
• Identify when a particle moves parallel to i or j or is at rest
• Use constant-acceleration equations in vector form
• Apply F = ma with vector forces
• Decide whether two particles meet and model a projectile with a position vector

Practice material written by Zestly, based on the DfE A level Mathematics subject content (kinematics: calculus for variable acceleration, extended to two dimensions using vectors; Newton's second law in vector form; projectiles).