Calculus in Kinematics

Position, Velocity, and Acceleration

In kinematics, calculus connects the three main quantities of motion:

Position function: \( x(t) \)
Velocity: first derivative of position → \( v(t) = \frac{dx}{dt} \)
Acceleration: first derivative of velocity → \( a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2} \)

\( v(t) = \frac{dx}{dt} \quad\quad a(t) = \frac{d^2x}{dt^2} \)

Using Integrals in Kinematics

We can also go in reverse using integrals:

From acceleration to velocity: \( v(t) = \int a(t)\,dt + C \)
From velocity to position: \( x(t) = \int v(t)\,dt + C \)

\( x(t) = \int \int a(t)\,dt^2 \)

Example 1: Derivative

If \( x(t) = 3t^2 + 2t \), then:

Velocity: \( v(t) = 6t + 2 \)
Acceleration: \( a(t) = 6 \)

Example 3: Conceptual

If you know the graph of \( x(t) \):

Slope = velocity
Curvature = acceleration (concave up = positive acceleration)

This position-time graph shows: increasing slope (increasing velocity) and concave up shape (positive acceleration)

Basic Calculus Rules

Here are some fundamental calculus rules useful for derivatives and integrals:

Power Rule (Derivative): \( \frac{d}{dx} x^n = nx^{n-1} \)
Reverse Power Rule (Integral): \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \)
Derivative of Natural Log: \( \frac{d}{dx} \ln|x| = \frac{1}{x} \)
Constant Rule: Derivative of a constant is zero: \( \frac{d}{dx} c = 0 \)
Constant Multiple Rule: \( \frac{d}{dx} [cf(x)] = c \frac{d}{dx} f(x) \)

\[ \begin{aligned} &\frac{d}{dx} x^n = nx^{n-1} \\ &\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \\ &\frac{d}{dx} \ln|x| = \frac{1}{x} \\ &\frac{d}{dx} c = 0 \\ &\frac{d}{dx} [cf(x)] = c \frac{d}{dx} f(x) \end{aligned} \]