$$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $\text{Extra close brace or missing open brace}$ $\text{Extra close brace or missing open brace}$ $$

Change in Velocity

When I write the Greek letter Δ (delta) in front of a variable, I am talking about the change in that variable, usually between some initial moment and some final moment. For instance, $\mathrm{\Delta }\overrightarrow{v}$ is the change in velocity, and $\mathrm{\Delta }t$ is the change in time. The change in a variable A is defined as

where $A_i$ is the variable's value at hte initial moment and $A_f$ is its value at the final moment. The change in $A$ is also the value you need to add to the initial value to get the final value:

The change in a vector quantity, like velocity, works exactly the same way: $$\mathrm{\Delta }\overrightarrow{v}={\overrightarrow{v}}_f-{\overrightarrow{v}}_i={\overrightarrow{v}}_f+(-{\overrightarrow{v}}_i)$$

Consider this motion diagram.

To calculate the change in velocity, we take the negative of ${\overrightarrow{v}}_i$ and add it to ${\overrightarrow{v}}_f$, to get $\mathrm{\Delta }\overrightarrow{v}$

$${\overrightarrow{v}}_f+(-{\overrightarrow{v}}_i)=\mathrm{\Delta }v$$

We can also think of $\mathrm{\Delta }v$ as what we add to ${\overrightarrow{v}}_i$ to get ${\overrightarrow{v}}_f$:

$${\overrightarrow{v}}_f={\overrightarrow{v}}_i+\mathrm{\Delta }\overrightarrow{v}$$

Remember that the displacement is written as $\mathrm{\Delta }\overrightarrow{x}$? While it is easiest to think of it as the vector from one spot to another, we can think of it as a "change of position". To do that, we define the position vector of a spot as the vector from the origin to that spot.