Kinematics
This unit builds the vocabulary the rest of the course leans on: position, velocity, and acceleration, and the precise difference between a vector and its magnitude. The kinematic equations, graphical analysis, free fall, and projectile motion are all the same handful of ideas applied to progressively less trivial situations -- master the sign conventions and the v-t/x-t graph relationships here, and Dynamics (Unit 2) and Circular Motion (Unit 3) stop being new content and start being "the same motion, now with a cause."
What you'll learn
- Interpret and construct position-time, velocity-time, and acceleration-time graphs for 1D motion.
- Calculate displacement, velocity, and acceleration using the kinematic equations for constant acceleration.
- Determine instantaneous velocity/acceleration from graph slopes and displacement from the area under a v-t graph.
- Analyze 2D projectile motion by separating it into independent horizontal and vertical components.
- Solve for range, maximum height, and time of flight in projectile motion.
- Distinguish average from instantaneous quantities.
- Apply relative velocity concepts across reference frames.
1. Describing Motion: Position, Displacement, and Distance
Kinematics describes how objects move without yet asking what causes the motion. Every problem starts the same way: pick a reference frame, draw a coordinate axis, and choose which direction is positive. That choice is arbitrary, but once made it must be used consistently for every quantity in the problem — a sign error from a flipped axis is one of the most common ways to lose points on an exam.
Position \(x\) is an object's location relative to the origin; it can be positive or negative. Displacement is the change in position, \(\Delta x = x_f - x_i\). Displacement is a vector: its sign tells you direction, and it depends only on the start and end points, not on the path taken. Distance, by contrast, is the total length of the path traveled — a scalar that is never negative and does depend on the path.
The distinction matters. If you jog 400 m east and then 100 m back west, the distance you covered is 500 m, but your displacement is only 300 m east. On a round trip that returns to the start, distance is large while displacement is exactly zero. Exam questions exploit this constantly, so read carefully whether a question asks "how far did it travel" (distance) or "how far from the start" (displacement).
2. Velocity and Speed
Average velocity is displacement over elapsed time, \(\bar v = \frac{\Delta x}{\Delta t}\). Because it is built from displacement, it is a vector and carries the same sign as the displacement. Average speed is a different quantity: total distance over elapsed time. It is a scalar and is never negative.
On any trip that changes direction, average speed and the magnitude of average velocity are not equal. For a round trip the average velocity is zero (displacement is zero) even though the average speed is clearly not — the object really was moving.
Instantaneous velocity is the value the average velocity approaches as the time interval shrinks toward zero; graphically it is the slope of the position-time graph at that instant. Speed is the magnitude of the instantaneous velocity. When a problem simply says "velocity" or "speed" without qualification, it almost always means the instantaneous value.
3. Acceleration
Acceleration is the rate of change of velocity: \(\bar a = \frac{\Delta v}{\Delta t} = \frac{v_f - v_0}{\Delta t}\), measured in m/s\(^2\). It is a vector. Instantaneous acceleration is the slope of the velocity-time graph at an instant.
The sign of the acceleration does not by itself tell you whether an object is speeding up or slowing down — that depends on the sign of the acceleration relative to the velocity. When acceleration and velocity point the same way, the object speeds up; when they point opposite ways, it slows down. An object moving in the negative direction with a negative acceleration is actually speeding up. The reliable question to ask is simply whether the speed (the magnitude of velocity) is growing or shrinking.
4. The Kinematic Equations (Constant Acceleration)
When acceleration is constant, four equations relate the five quantities \(\Delta x\), \(v_0\), \(v\), \(a\), and \(t\). Each equation is missing exactly one of them, which is the key to choosing the right one:
\[ v = v_0 + at \qquad (\text{missing } \Delta x) \]
\[ \Delta x = v_0 t + \tfrac{1}{2}a t^2 \qquad (\text{missing } v) \]
\[ v^2 = v_0^2 + 2a\,\Delta x \qquad (\text{missing } t) \]
\[ \Delta x = \tfrac{1}{2}(v_0 + v)\,t \qquad (\text{missing } a) \]
These are not four unrelated facts. The first comes straight from the definition of constant acceleration. Because velocity changes linearly when \(a\) is constant, the average velocity is simply \(\tfrac{1}{2}(v_0 + v)\); multiplying by \(t\) gives the fourth equation. Substituting the first into the fourth gives the second, and eliminating \(t\) between them gives the third.
Problem-solving strategy: list what you know and what you want, then pick the equation that omits the one variable you neither have nor need. Assign signs consistently with your chosen axis. Remember these equations are valid only while acceleration is constant — if it changes partway through, split the motion into segments and solve each separately.
5. Graphical Analysis of Motion
Graphs encode motion, and reading them is a guaranteed exam skill. On a position-time graph, the slope is the velocity: a straight sloped line means constant velocity, a curve means changing velocity (acceleration), and a horizontal line means the object is at rest.
On a velocity-time graph, the slope is the acceleration, and the area between the line and the time axis is the displacement. On an acceleration-time graph, the area between the line and the time axis is the change in velocity. Areas below the axis count as negative.
To pull numbers off a graph, compute slopes as rise over run and areas by breaking the region into triangles and rectangles. This lets you solve many problems without ever plugging into an equation.
6. Free Fall
Near Earth's surface, and neglecting air resistance, every object in free fall accelerates downward at \(g \approx 9.8\) m/s\(^2\) (often rounded to 10 for quick estimates), regardless of its mass. "Free fall" means gravity is the only force acting — which includes objects thrown upward, since they remain in free fall the entire time after release.
Choose up as positive, so \(a = -g\), and the kinematic equations apply directly. At the highest point of an upward throw the velocity is instantaneously zero, but the acceleration is still \(-g\); gravity never pauses. The trajectory is symmetric about the peak: the time up equals the time down, and the speed passing any height on the way up equals the speed passing it on the way down.
7. Projectile Motion
A projectile moves under gravity alone after launch. The single most important idea is that horizontal and vertical motion are completely independent and share only one thing: the time \(t\). Treat them as two separate 1D problems linked by that common time.
Horizontally there is no acceleration (ignoring air resistance), so the horizontal velocity is constant, \(v_x = v_{0x}\), and \(x = v_{0x}\,t\). Vertically the motion is free fall, so \(v_y = v_{0y} - g t\) and \(y = v_{0y}\,t - \tfrac{1}{2}g t^2\).
For a launch at speed \(v_0\) and angle \(\theta\) above the horizontal, resolve into components: \(v_{0x} = v_0\cos\theta\) and \(v_{0y} = v_0\sin\theta\). When the launch and landing heights are the same on level ground, the time of flight is \(t = \frac{2v_{0y}}{g}\), the range is \(R = v_{0x}\,t\), and the maximum height is \(h = \frac{v_{0y}^2}{2g}\).
8. Relative Motion
Every velocity is measured relative to some reference frame. To convert between frames, add velocity vectors: the velocity of A relative to C equals the velocity of A relative to B plus the velocity of B relative to C, \(\vec v_{AC} = \vec v_{AB} + \vec v_{BC}\). In one dimension this is ordinary signed addition; in two dimensions it is full vector addition.
A classic two-dimensional case is crossing a current or flying through wind: the object's velocity relative to the ground equals its velocity relative to the water (or air) plus the water's (or air's) velocity relative to the ground. When the two velocities are perpendicular, combine them with the Pythagorean theorem for the magnitude and a tangent for the direction.
Key equations
- v = v0 + at — Constant acceleration, displacement not needed
- x = x0 + v0t + 1/2at^2 — Solving for position over time
- v^2 = v0^2 + 2a(x - x0) — Time is unknown or not needed
- x = x0 + 1/2(v0 + v)t — Average-velocity form, when a isn't given directly
- R = (v0^2 sin2θ)/g — Range when launch/landing heights are equal