Definition: Speed and velocity

The speed of a particle is measured in terms of the total distance $d$ traveled in time $\Delta t$, which is a scalar (only carries magnitude) and is always a positive value: $$s=\frac d{\Delta t}$$ On the other hand, velocity is a vector, meaning it also has a direction. Velocity is measured using the change in position (displacement) over time (it doesn't care about the path you take). $$v=\frac{\Delta x}{\Delta t}$$ When you define a velocity in 1-D, you must therefore define a direction to be positive velocity (and displacement), and a direction to be negative velocity (and displacement).

Definition: Reference frames

What the velocity of an object is depends on its reference frame. The Earth is a reference frame -- if we take the Earth as a reference frame, then anything at rest with respect to the Earth has velocity 0. However, if we have a moving train, we can take the train as a reference frame. Everything at rest with respect to the train is in fact moving with respect to the Earth.

Definition: Acceleration

The acceleration of an object (a vector) is defined as the change in velocity over time. We define the positive and negative directions of acceleration to be the same as those of velocity and displacement.$$a=\frac{\Delta v}{\Delta t}$$

Theorem: Graphs

When we have graphs of position, velocity, and acceleration with respect to time, we can extract information from a graph about the other quantities involved.

• Slopes: In an $x-t$ graph, the slope at some point is the velocity at that given moment. In a $v-t$ graph, the slope at some point is the acceleration at that given moment.
• Areas under graphs: In a $v-t$ graph, the area under the graph between two points is the displacement between those two points. In an $a-t$ graph, the area under the graph between two points is the change in velocity between those two points.

Theorem: Kinematic equations

For motion with constant acceleration, we have the following two equations governing kinematic motion: $$\Delta x=v_0t+\frac12at^2\qquad v_f^2-v_0^2=2a\Delta x$$ It's crucial to memorize these! Make sure you also have the signs (positive versus negative) of all the quantities consistent with which direction you define as positive, and which you define as negative.