Using Calculus in 2D

• 2D Vector Kinematics: In two dimensions, motion is described using position vectors $\mathbf{r} = x\mathbf{i} + y\mathbf{j}$, velocity vectors $\mathbf{v} = v_x\mathbf{i} + v_y\mathbf{j}$, and acceleration vectors $\mathbf{a} = a_x\mathbf{i} + a_y\mathbf{j}$.

• Component Independence: The fundamental principle of vector calculus is that the $i$ and $j$ components are treated as independent functions of time ($t$), allowing scalar calculus rules to be applied to each direction separately.

• Scalar vs. Vector Quantities: It is vital to distinguish between vectors (like velocity) and their magnitudes (like speed). While time is a scalar, displacement, velocity, and acceleration are all vectors characterized by both magnitude and direction.

• Calculus Relationship: The relationship between kinematics vectors is identical to the 1D case but applied to the entire vector: $\mathbf{v} = \frac{d\mathbf{r}}{dt}$ and $\mathbf{a} = \frac{d\mathbf{v}}{dt}$.

• Linearity of Differentiation: Because the derivative of a sum is the sum of the derivatives, differentiating $\mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j}$ yields $\frac{dx}{dt}\mathbf{i} + \frac{dy}{dt}\mathbf{j}$.

• Fundamental Theorem in 2D: Integration recovers the vector quantities up to a constant vector: $\mathbf{r} = \int \mathbf{v}\, dt + \mathbf{c}$, where $\mathbf{c}$ represents the initial position at $t=0$ if the integral starts from zero.

Differentiating Vectors

• Process: Apply standard differentiation rules (Power, Product, Chain, etc.) to each component function independently.
• Result: The resulting vector represents the rate of change in both magnitude and direction of the original vector.

Integrating Vectors

• Step 1: Integrate each component separately to find the indefinite integral.
• Step 2: Add a constant vector $\mathbf{c} = c_1\mathbf{i} + c_2\mathbf{j}$ to the result.
• Step 3: Use boundary conditions (initial values) to solve for $c_1$ and $c_2$ by equating components.

Finding Magnitudes

• Speed: To find speed from velocity, calculate the magnitude of the velocity vector using the Pythagorean theorem: $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}$.

• Displacement vs. Position: Displacement ($\mathbf{s}$) is the change in position, while the position vector ($\mathbf{r}$) is the coordinate relative to a fixed origin, defined as $\mathbf{r} = \mathbf{r}_0 + \mathbf{s}$.

• Distance vs. Magnitude of Displacement: Distance is the total path length traveled, while the magnitude of displacement is the straight-line length from the start to the current position.

• The Forgotten Constant: Forgetting the constant vector $\mathbf{c}$ during integration leads to assuming the particle starts at the origin when it may not.

• Mixing Components: Incorrectly integrating the $i$ component with respect to $y$ or vice versa. Always maintain strictly separate treatment of $i$ and $j$ until the very end if magnitude is needed.

• SUVAT Misuse: Attempting to use SUVAT equations when the acceleration is a function of time. SUVAT is only valid if the acceleration vector is constant (e.g., $\mathbf{a} = 2\mathbf{i} - 3\mathbf{j}$).