218 Velocity
A vector rate of change of position that is used to describe how fast and in which direction an object moves, where a rate of change is a derivative with respect to time.
Average velocity. Average velocity is the net displacement divided by the elapsed time. Displacement is the change of position from start to finish. This principle is used to describe overall progress over a finite interval.
The average velocity is
\[ \mathbf{v}_{\mathrm{avg}} = \dfrac{\Delta\mathbf{r}}{\Delta t} \]
where
- \(\mathbf{v}_{\mathrm{avg}}\) is the average velocity.
- \(\Delta\mathbf{r}\) is the displacement.
- \(\Delta t\) is the elapsed time.
Instantaneous velocity. Instantaneous velocity is the limit of the average velocity as the time interval shrinks to zero. This principle is used to give the speed and direction at one moment.
The instantaneous velocity is
\[ \mathbf{v} = \lim_{\Delta t \to 0}\dfrac{\Delta\mathbf{r}}{\Delta t} \]
where
- \(\mathbf{v}\) is the instantaneous velocity.
- \(\Delta\mathbf{r}\) is the displacement.
- \(\Delta t\) is the elapsed time.
Velocity as the time derivative of position. Velocity is the time derivative of the position vector and points along the path of motion. This principle is used to read the three components of motion from the coordinates.
The velocity as a derivative is
\[ \mathbf{v} = \dfrac{d\mathbf{r}}{dt} \]
The Cartesian components are
\[ \mathbf{v} = \dfrac{dx}{dt}\,\hat{\mathbf{i}} + \dfrac{dy}{dt}\,\hat{\mathbf{j}} + \dfrac{dz}{dt}\,\hat{\mathbf{k}} \]
where
- \(\mathbf{v}\) is the velocity.
- \(\mathbf{r}\) is the position vector.
- \(t\) is time.
- \(x\), \(y\), and \(z\) are the Cartesian coordinates.
Relative velocity. The relative velocity of one object with respect to another is the difference of their velocities. This principle is used to describe motion as seen from a moving observer.
The relative velocity is
\[ \mathbf{v}_{CA} = \mathbf{v}_{C} - \mathbf{v}_{A} \]
where
- \(\mathbf{v}_{CA}\) is the velocity of \(C\) relative to \(A\).
- \(\mathbf{v}_{C}\) is the velocity of \(C\).
- \(\mathbf{v}_{A}\) is the velocity of \(A\).
Uniform motion. Uniform motion is motion at constant velocity. This principle is used to identify the case of vanishing acceleration and vanishing net force.
Uniform motion is
\[ \mathbf{v} = \mathrm{constant} \implies \mathbf{a} = 0 \]
where
- \(\mathbf{v}\) is the velocity.
- \(\mathbf{a}\) is the acceleration.
Note: Also called the velocity vector.
218.1 References
- Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. §1.4–2.2 — average and instantaneous velocity.
- Boas, M. L. Mathematical Methods in the Physical Sciences. 3rd ed. Wiley, 2005. — \(\mathbf{v}=\dfrac{d\mathbf{r}}{dt}\) with Cartesian components.
- Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. §1.1 — \(\mathbf{v}=\lim_{\Delta t\rightarrow 0}\dfrac{\Delta\mathbf{r}}{\Delta t}=\dfrac{d\mathbf{r}}{dt}\).
- Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics, Vol. I. Ch. 8 — vector velocity as \(\mathbf{v}=\dfrac{d\mathbf{r}}{dt}\).