Calculus-Based Kinematics Cheat Sheet

The core ideas of Calculus-Based Kinematics distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Instantaneous velocity is the time derivative of position: (t) = rac{dx}{dt}$.

Acceleration is (t) = rac{dv}{dt} = rac{d^2x}{dt^2}$.

Given acceleration, integrate to get velocity, integrate again to get position. Each step needs an initial condition.

By the chain rule, = v rac{dv}{dx}$. Useful when acceleration depends on position rather than time.

Displacement is the signed integral of velocity. Distance is the integral of the absolute value of velocity.

When acceleration varies, standard kinematic equations fail. Use calculus: set up the ODE and integrate.

Slope of x-t graph = velocity. Slope of v-t graph = acceleration. Area under v-t graph = displacement.

Many problems reduce to separable ODEs. Separate variables and integrate both sides independently.

Instantaneous Velocity:The derivative of position with respect to time: v(t) = dx/dt. Measured in m/s.

Instantaneous Acceleration:The derivative of velocity with respect to time: a(t) = dv/dt = d^2x/dt^2. Measured in m/s^2.

Jerk:The derivative of acceleration: j(t) = da/dt = d^3x/dt^3. Measured in m/s^3.

Displacement:The signed change in position: integral of v(t) dt over a time interval. Can be positive, negative, or zero.

Distance:The total path length traveled: integral of |v(t)| dt. Always non-negative.

Initial Condition:A known value of position, velocity, or acceleration at a specific time, used to determine constants of integration.

Separable ODE:A differential equation that can be written as f(v) dv = g(t) dt, allowing each side to be integrated independently.

Chain Rule Form:The relation a = v dv/dx, derived from a = dv/dt and v = dx/dt via the chain rule. Used for position-dependent acceleration.

Terminal Velocity:The constant velocity reached when the drag force equals the driving force (e.g., gravity). Found by setting a = 0 in the ODE.

Power Rule:Differentiation: d/dt(t^n) = nt^{n-1}. Integration: integral of t^n dt = t^{n+1}/(n+1) + C.

Constant of Integration:An arbitrary constant C introduced with each indefinite integral, determined by boundary or initial conditions.

Non-Constant Acceleration:Acceleration that varies with time, velocity, or position. Requires calculus (not kinematic equations) to solve.

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