suvat in 2D

Four of the five standard kinematic equations can be written directly in vector form because they involve linear combinations of vectors and scalars.

Velocity-Time Relation: $\mathbf{v} = \mathbf{u} + \mathbf{a}t$. This shows that the final velocity is the vector sum of the initial velocity and the change in velocity caused by acceleration over time.

Displacement-Velocity Relation: $\mathbf{s} = \frac{1}{2}(\mathbf{u} + \mathbf{v})t$. This calculates displacement based on the average velocity vector over the time interval.

Displacement-Acceleration Relations: $\mathbf{s} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2$ and $\mathbf{s} = \mathbf{v}t - \frac{1}{2}\mathbf{a}t^2$. These are the most common equations used to find position when acceleration is known.

Component Method: You can split a 2D problem into two independent 1D problems—one for the horizontal ($\mathbf{i}$) direction and one for the vertical ($\mathbf{j}$) direction. Solve for the unknown in one dimension and use the result (often time $t$) to solve the other.

Vector Equation Method: Alternatively, you can substitute the $\mathbf{i}$ and $\mathbf{j}$ components directly into the vector formulas. This often leads to a system of two simultaneous equations (one for each component).

Magnitude Calculations: To find the speed of an object, calculate the magnitude of the velocity vector using Pythagoras: $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}$. Similarly, the distance from the start is the magnitude of the displacement vector: $|\mathbf{s}| = \sqrt{s_x^2 + s_y^2}$.

Check for Constant Acceleration: Always verify that the acceleration is a constant vector (e.g., $2\mathbf{i} - 3\mathbf{j}$) and not a function of time (e.g., $2t\mathbf{i}$). If it depends on $t$, SUVAT is not applicable.

Parallel Motion: If a question states a particle is moving 'parallel to' a vector like $2\mathbf{i} + \mathbf{j}$, its velocity must be a scalar multiple of that vector: $\mathbf{v} = k(2\mathbf{i} + \mathbf{j})$.

Zero Components: If an object is moving 'due North' or 'parallel to the $\mathbf{j}$ vector', the $\mathbf{i}$ component of its velocity is zero. Use this to set up and solve equations for unknown variables like time.

Sanity Check: Ensure your time $t$ is always positive. If a quadratic in $t$ yields a negative and a positive result, the negative one is physically impossible in standard kinematics.

Mixing Components: A common error is using the horizontal component of initial velocity ($u_x$) with the vertical component of acceleration ($a_y$) in the same equation. Components must remain strictly separated unless using full vector notation.

Distance vs. Displacement: Displacement is the vector from the start point to the end point. Distance is the magnitude of that vector. Students often confuse 'distance travelled' (path length) with 'distance from origin' (magnitude of displacement).

Units and Vectors: When asked for 'acceleration', provide the vector $(a\mathbf{i} + b\mathbf{j})$ unless the question specifically asks for the 'magnitude of acceleration'.