Velocity, Acceleration

Velocity

• Merriam-Webster dictionary: "quickness of motion; rapidity of movement", "the rate of change of position along a straight line with respect to time : the derivative of position with respect to time"
• Google dictionary: "the speed of something in a given direction."
• Wikipedia: "Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time."

In physics, velocity is represented by a vector $\vec{v}$, with a direction and a magnitude. Its magnitude, short-handed as $||\vec{v}||$, is called speed. In the SI unit system, the unit of velocity and the unit of speed are both $m/s$.

• Average velocity: $\overline{\vec{v}}=\frac{\Delta \vec{x}}{\Delta t}$
• Instantaneous velocity: $\vec{v}=\lim\limits_{\Delta t\to 0}\frac{\Delta \vec{x}}{\Delta t}$

where $\vec{x}$ represents the position of the object, $\Delta$ means the difference of a physical quantity, specifically always the final value minus the initial value; $\lim\limits_{\Delta t\to 0}$ means when the time difference is infinitesimal small.

Simply put, velocity is the rate of change of position with respect to time. Average velocity considers an arbitrarily long period of time, and instantaneous velocity considers an extremely short period of time.

If a velocity has a negative value, it means that you have defined a direction as the "positive direction", and the opposite of that direction is negative.

Acceleration

The definition of acceleration in daily life:

• Merriam-Webster dictionary: the act or process of moving faster or happening more quickly
• Google dictionary: a vehicle's capacity to gain speed within a short time

The definition of acceleration in physics:

Merriam-Webster dictionary: the rate of change of velocity with respect to time Google dictionary: the rate of change of velocity per unit of time Wikipedia: the rate of change of the velocity of an object with respect to time.

Acceleration is also represented by a vector $\vec{a}$, with a direction and a magnitude. In the SI unit system, the unit of acceleration is $m/s^2$.

• Average acceleration: $\overline{\vec{a}}=\frac{\Delta \vec{v}}{\Delta t}$
• Instantaneous acceleration: $\vec{a}=\lim\limits_{\Delta t\to 0}\frac{\Delta \vec{v}}{\Delta t}$ where $\vec{v}$ represents the velocity of the object, $\Delta$ means the difference of a physical quantity, specifically always the final value minus the initial value; $\lim\limits_{\Delta t\to 0}$ means when the time difference is infinitesimal small.

If an acceleration has a negative value, it means that you have defined a direction as the "position direction", and the opposite of that direction is negative. There are multiple cases:

Velocity	Acceleration	Meaning
+	+	move to the positive direction and increase speed
+	-	move to the positive direction and decrease speed
-	-	move to the negative direction and increase speed
-	+	move to the negative direction and decrease speed