Kinematics

A scalar has only a size (like distance, speed or time), while a vector has a size and a direction (like position, displacement, velocity and acceleration). You can write a vector as a magnitude with a direction or in unit vector notation, such as $\vec{r} = 3\hat{i} + 4\hat{j}$, and you add vectors by adding their components.

Displacement is your change in position, and average velocity and average acceleration divide a change by the time it took. Shrink that time interval toward zero and you get instantaneous values, which are derivatives: $v_x = \frac{dx}{dt}$ and $a_x = \frac{dv_x}{dt}$. An object is accelerating whenever its speed or its direction of motion changes.

Motion can be shown with diagrams, graphs, equations and words, and you should be able to move between them. The slope of a position–time graph is velocity, and the slope of a velocity–time graph is acceleration. The area under a velocity–time graph is the displacement, $\Delta x = \int v_x\,dt$, and the area under an acceleration–time graph is the change in velocity. The kinematic equations, such as $x = x_0 + v_0t + \frac{1}{2}at^2$, work only when acceleration is constant, like free fall near Earth's surface.

Every measurement of motion depends on the observer's reference frame. To switch frames you add or subtract velocity vectors: an object's velocity relative to the ground is its velocity relative to a moving frame plus that frame's velocity relative to the ground. Observers in different inertial (non-accelerating) frames disagree about velocity but measure the same acceleration.

You can split motion in two dimensions into perpendicular components. Each component follows its own one-dimensional kinematics, and what happens in one direction doesn't change the other. In projectile motion (ignoring air resistance), the horizontal velocity stays constant while the vertical acceleration is g downward. When position is given as a function of time, you differentiate each component to get the velocity and acceleration vectors.