Kinematics

A scalar has only a size (like distance, speed, time or mass), while a vector has a size and a direction (like displacement, velocity or acceleration). In one dimension you show direction with a sign: choose which way is positive, give anything pointing the other way a negative sign, and adding vectors becomes adding signed numbers.

Displacement is your change in position (Δx = x_f − x_i), not the total distance you traveled. Average velocity is displacement divided by the time it took, and average acceleration is how much the velocity changes per second (Δv/Δt). You're accelerating whenever your speed or direction changes, and over a very short time interval these averages become the instantaneous values.

You can show the same motion with motion diagrams, graphs, equations or words. On a position–time graph the slope is velocity; on a velocity–time graph the slope is acceleration and the area under the curve is displacement; and the area under an acceleration–time graph is the change in velocity. The three kinematic equations work only when acceleration is constant, as in free fall near Earth, where the acceleration is g ≈ 9.8 m/s² downward (AP problems often round it to 10 m/s²).

What you measure depends on your reference frame: a passenger sitting on a moving train is at rest relative to the train but moving relative to the ground. To switch frames in one dimension you add or subtract velocities (a ball's velocity relative to the ground = its velocity relative to the train + the train's velocity relative to the ground), and every inertial observer measures the same acceleration.

Any vector can be split into perpendicular x and y components using sine, cosine and the Pythagorean theorem, and two-dimensional motion can then be solved as two one-dimensional problems that share the same time. In projectile motion with no air resistance, the horizontal velocity stays constant while the vertical motion has a constant downward acceleration g.