5.4: Velocity

Velocity is displacement per unit of time. The average velocity of point mass \(i\) between time \(t_{1}\) and \(t_{2}\), over path curve \(\overrightarrow{\boldsymbol{r}}_{s}\), over a time interval \(\Delta t=t_{2}-t_{1}\) is:

\[v_{s, \text { avg }}=\frac{\Delta s_{i, 12}}{\Delta t} \tag{5.18} \label{5.18}\]

This average velocity can be positive or negative. In contrast, the average speed, is defined to be always positive:

\[v_{s, \mathrm{sp}, \mathrm{avg}}=\frac{\Delta s_{\mathrm{T}, i, 12}}{\Delta t} \tag{5.19} \label{5.19}\]

The average velocity and speed are determined over a large time interval, in contrast the instantaneous velocity is measured over an infinitesimally small time interval.

Concept: Velocity Vector

The (instantaneous) velocity vector \(\overrightarrow{\boldsymbol{v}}_{i}\) is defined as the time derivative of the position vector:

\[\overrightarrow{\boldsymbol{v}}_{i}(t) \equiv \frac{\mathrm{d} \overrightarrow{\boldsymbol{r}}_{i}}{\mathrm{~d} t} \equiv \lim _{\mathrm{d} t \rightarrow 0} \frac{\overrightarrow{\boldsymbol{r}}_{i}(t+\mathrm{d} t)-\overrightarrow{\boldsymbol{r}}_{i}(t)}{\mathrm{d} t} \tag{5.20} \label{5.20}\]

The determination of the velocity vector from the position vector is shown graphically in Figure 5.5, taking a relatively large timestep \(\mathrm{d} t=1 \mathrm{~s}\). The speed or absolute velocity \(v_{i}(t)\) is defined as the magnitude of the velocity vector, \(v_{i}(t)=\left|\overrightarrow{\boldsymbol{v}}_{i}(t)\right|\) which is always positive or zero.

Derivation: Velocity on the path curve

If a path curve has been defined, it can be shown that the velocity vector is always tangential to this path curve:

\[\overrightarrow{\boldsymbol{v}}_{i}=\frac{\mathrm{d} \overrightarrow{\boldsymbol{r}}_{i}}{\mathrm{~d} t}=\frac{\mathrm{d} \overrightarrow{\boldsymbol{r}}_{s}\left(s_{i}(t)\right)}{\mathrm{d} t}=\frac{\mathrm{d} s_{i}}{\mathrm{~d} t} \frac{\mathrm{d} \overrightarrow{\boldsymbol{r}}_{s}(s)}{\mathrm{d} s}=v_{s, i} \hat{\boldsymbol{s}} \tag{5.21} \label{5.21}\]

To derive this equation, we used the chain rule and used that the vector \(\frac{\mathrm{d} \overrightarrow{\boldsymbol{r}}_{s}(s)}{\mathrm{d} s}\) has a magnitude of 1 , since \(|\mathrm{d} s|=|\mathrm{d} \boldsymbol{\boldsymbol { r }}|\) (see previous section). As can be seen in Equation 5.21, we define \(\hat{\boldsymbol{s}} \equiv \frac{\mathrm{d} \overrightarrow{\boldsymbol{r}}_{s}(s)}{\mathrm{d} s}\) as the unit vector tangential to the path curve that points in the direction of the path curve in which \(s\) increases (see Figure 5.2) and we define the path velocity \(v_{s}\) as:

\[v_{s, i} \equiv \frac{\mathrm{d} s_{i}}{\mathrm{~d} t}=\dot{s}_{i} \tag{5.22} \label{5.22}\]

Figure 5.5: Based on the motion of the point mass in Figure 5.1, the velocity vectors \(\overrightarrow{\boldsymbol{v}}_{i}(t)\) are determined at every time \(t\), using a time-step \(\mathrm{d} t=1 \mathrm{~s}\). On the right the acceleration vectors \(\overrightarrow{\boldsymbol{a}}_{i}(t)\) are determined from \(\overrightarrow{\boldsymbol{v}}_{i}(t)\) in the same way as the velocity vectors were determined from the position vectors. Note that the average velocity vectors between the points with \(\Delta t=1 \mathrm{~s}\) are drawn.

The subscript \(s\) indicates that \(v_{s, i}\) is the velocity of object \(i\) along the path \(\overrightarrow{\boldsymbol{r}}_{s}\). Equation (5.21) proves that the velocity vector of the point mass \(i\) is always parallel (tangential) to the path curve unit vector \(\hat{\boldsymbol{s}} . v_{s, i}\) can be both positive and negative, depending whether it is in the same or the opposite direction as the positive \(s\) direction \(\hat{\boldsymbol{s}}\) and its magnitude is equal to that of the velocity vector \(v_{i} \equiv\left|\overrightarrow{\boldsymbol{v}}_{i}\right|=\left|v_{s, i}\right|\).