Using Calculus in 1D Kinematics

Kinematic Variables as Functions: In 1D motion, displacement ($s$), velocity ($v$), and acceleration ($a$) are expressed as functions of time ($t$). This allows for the analysis of 'variable acceleration' scenarios where standard SUVAT equations do not apply.

Velocity ($v$): Defined as the instantaneous rate of change of displacement with respect to time, represented mathematically as $v = \frac{ds}{dt}$.

Acceleration ($a$): Defined as the instantaneous rate of change of velocity with respect to time, represented as $a = \frac{dv}{dt}$ or the second derivative of displacement $\frac{d^2s}{dt^2}$.

Finding Velocity and Acceleration: To find $v(t)$ from $s(t)$, or $a(t)$ from $v(t)$, apply standard differentiation rules to the function. This provides an expression for the rate of change at any instant $t$.

Indefinite Integration for General Forms: To find $v(t)$ from $a(t)$, use $\int a(t) \, dt = v(t) + c$. The constant $c$ is determined by using 'initial conditions' (e.g., the velocity at $t=0$).

Definite Integration for Changes: To find the change in position between $t_1$ and $t_2$, evaluate $\int_{t_1}^{t_2} v(t) \, dt$. This yields the displacement directly without needing to solve for the constant $c$ explicitly.

Vector vs. Scalar: Velocity and acceleration are vectors, meaning they can be negative to indicate direction. Speed and distance are scalars and are always non-negative.

Initial vs. Starting from Rest: 'Initially' refers to the state at $t=0$. 'Starting from rest' specifically implies that $v=0$ when $t=0$.

Check for Direction Changes: When asked for 'total distance', always check if the velocity function $v(t)$ crosses the x-axis (becomes zero) within the given time interval. If it does, you must integrate the sections above and below the axis separately and sum their absolute values.

Boundary Conditions: Always look for phrases like 'at the origin' ($s=0$) or 'initially' ($t=0$) to find the constants of integration. Forgetting the $+c$ is the most common way to lose marks in kinematics problems.

Sanity Checks: Ensure the units are consistent (e.g., $m$, $ms^{-1}$, $ms^{-2}$). If a particle is 'moving to the left' or 'decelerating', ensure your signs reflect the chosen coordinate system.

SUVAT Misuse: Students often try to use $v = u + at$ when acceleration is a function of time (e.g., $a = 3t^2$). These equations only work for constant acceleration; calculus is mandatory for variable acceleration.

Displacement vs. Position: Integrating velocity gives the change in displacement. If the particle did not start at the origin ($s \neq 0$ at $t=0$), the displacement function must include the initial position constant.

Differentiating instead of Integrating: In the heat of an exam, students sometimes differentiate $v$ to find $s$ instead of integrating. Remember: $s \to v \to a$ is 'down' (differentiation), $a \to v \to s$ is 'up' (integration).