5.5: Acceleration
Similarly to the way the velocity vector is determined from the position vector, the acceleration vector can be determined from the velocity vector (see Figure 5.5).
The (instantaneous) acceleration vector \(\overrightarrow{\boldsymbol{a}}_{i}\) is defined as the time derivative of the velocity vector:
\[\overrightarrow{\boldsymbol{a}}_{i}(t) \equiv \frac{\mathrm{d} \overrightarrow{\boldsymbol{v}}_{i}}{\mathrm{~d} t} \equiv \lim _{\mathrm{d} t \rightarrow 0} \frac{\overrightarrow{\boldsymbol{v}}_{i}(t+\mathrm{d} t)-\overrightarrow{\boldsymbol{v}}_{i}(t)}{\mathrm{d} t} \tag{5.23} \label{5.23}\]
The absolute acceleration is defined as: \(a_{i}=\left|\overrightarrow{\boldsymbol{a}}_{i}\right|\). By taking the time derivative of equation (5.21) using the product rule it is found that:
\[\overrightarrow{\boldsymbol{a}}_{i}(t)=\frac{\mathrm{d} \overrightarrow{\boldsymbol{v}}_{i}}{\mathrm{~d} t}=\frac{\mathrm{d} v_{s, i}}{\mathrm{~d} t} \hat{\boldsymbol{s}}+v_{s, i} \frac{\mathrm{d} \hat{\boldsymbol{s}}}{\mathrm{d} t} \tag{5.24} \label{5.24}\]
In contrast to the velocity vector, the acceleration vector is not tangential to the path curve \(\hat{\boldsymbol{s}}\), except if the path is straight \(\left(\frac{\mathrm{d} \hat{\boldsymbol{s}}}{\mathrm{d} t}=\overrightarrow{\mathbf{0}}\right)\), or if the speed is zero. Since the magnitude of \(\hat{\boldsymbol{s}}\) is constant, the vector \(\frac{\mathrm{d} \hat{\boldsymbol{s}}}{\mathrm{d} t}\) is perpendicular to \(\hat{\boldsymbol{s}}\). For that reason we have for the acceleration component tangential to the path curve \(a_{i, s, t}\) :
\[a_{s, i, t}=\overrightarrow{\boldsymbol{a}}_{i} \cdot \hat{\boldsymbol{s}}=\frac{\mathrm{d} v_{s, i}}{\mathrm{~d} t} \tag{5.25} \label{5.25}\]
This tangential component of the acceleration vector is also called the path acceleration \(a_{s, i}\) and can be different from the absolute value of the acceleration vector \(\left(\left|a_{s, i}\right| \neq\left|\overrightarrow{\boldsymbol{a}}_{i}\right|\right)\). We will discuss later how the acceleration vector in various coordinate systems can be determined, which will be very important for applying Newton’s second law.