Physics C: Mechanics · Unit 1: Kinematics · 16 min read · Updated 2026-05-11
Calculus-based Kinematics — AP Physics C: Mechanics
AP Physics C: Mechanics · Unit 1: Kinematics · 16 min read
1. Derivative Definitions of Velocity and Acceleration★★☆☆☆⏱ 3 min
Unlike algebra-based kinematics which only handles constant acceleration, calculus-based kinematics describes motion with changing acceleration using instantaneous rates of change. Position $x(t)$ gives an object's location along a 1D axis at time $t$.
Exam tip: Examiners regularly test that you distinguish instantaneous velocity (derivative) from average velocity ($\Delta x/\Delta t$), which are only equal if acceleration is constant.
2. Integration to Find Velocity and Position★★☆☆☆⏱ 4 min
Integration reverses differentiation, allowing you to recover velocity from a known acceleration function, and position from a known velocity function, even when acceleration is not constant. This relies on using initial conditions to find constants of integration.
Starting from the definition of acceleration, we rearrange and integrate to get the general form for velocity:
v(t) = v_0 + \int_{0}^{t} a(t')dt'
Similarly, rearranging the velocity definition and integrating gives position, where $v_0$ is initial velocity at $t=0$, and $x_0$ is initial position at $t=0$:
x(t) = x_0 + \int_{0}^{t} v(t')dt'
Exam tip: Never forget constants of integration! They correspond to initial conditions, which are always provided in exam problems, and forgetting them costs 1-2 marks per question.
3. Variable Acceleration Motion★★★☆☆⏱ 5 min
A key difference between AP Physics 1 and AP Physics C is that AP Physics C regularly tests motion with non-constant acceleration. Constant-acceleration kinematic equations are only valid for constant acceleration, so you must use calculus for all other cases. When acceleration is given as a function of position $a(x)$, we use the chain rule to rewrite acceleration:
a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = v\frac{dv}{dx}
Rearranging gives a separable equation that can be integrated with the given initial and final bounds: $v dv = a(x) dx$
4. Calculus-Based Projectile Motion★★★★☆⏱ 3 min
Projectile motion is always separated into independent horizontal ($x$) and vertical ($y$) components, even when acceleration is non-constant. For ideal projectile motion (no air resistance), acceleration is constant ($a_x=0$, $a_y=-g$) and integration gives the familiar algebra-based results. For projectile motion with air resistance proportional to velocity, acceleration is non-constant and requires integration to solve.
5. 2D Vector Kinematics★★☆☆☆⏱ 3 min
All kinematic quantities are vectors in 2D, so we differentiate and integrate each vector component independently. We typically use unit vectors $\hat{i}$ (positive x-direction) and $\hat{j}$ (positive y-direction) to write vector expressions.
Position vector: $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$