Some of the content of this guide was modeled after a guide published by OpenStax and has been adapted for the NWP Learning Commons in March 2022. This work is licensed under a Creative Commons BY 4.0 International License.
For all motion problems, a reference frame needs to be determined. It is assumed in the previous examples that all of the motion is relative to Earth, but even the Earth moves relative to the Sun, and the Sun moves relative to the Milky Way. A reference frame is typically the larger body that carries a smaller body. For example, if a person is walking on a train, they have a velocity relative to the train itself, but the train has a velocity relative to the Earth. This in turn means the person has a different velocity relative to the Earth!
$$\vec{\textbf{v}}_{P\to{E}}=\vec{\textbf{v}}_{P\to{T}}+\vec{\textbf{v}}_{T\to{E}}$$
In the above equation, the smaller body is in relation to the larger body.
$$P\to{E}-Person\,to\,Earth,\quad{P\to{T}}-Person\,to\,Train,\quad{T\to{E}}-Train\,to\,Earth$$
$$\vec{\textbf{v}}_{P\to{E}}=\vec{\textbf{v}}_{P\to{T}}+\vec{\textbf{v}}_{T\to{E}}$$
$$\vec{\textbf{v}}_{P\to{E}}=-2{km}/{hr}+80{km}/{hr}$$
$$\vec{\textbf{v}}_{P\to{E}}=78{km}/{hr}$$