8.1: Principle of impulse and momentum
We first define the concepts impulse and momentum, then introduce the principle of impulse and momentum and finally present the derivation of it. These concepts are also illustrated in Figure 8.1.
The impulse vector \(\overrightarrow{\boldsymbol{J}}_{i, 12}\) is defined as the time integral of the sum of all forces \(j\) acting on a mass \(m_{i}\) over a specific time interval \(t_{1}-t_{2}\) :
\[\overrightarrow{\boldsymbol{J}}_{i, 12} \equiv \sum_{j} \int_{t_{1}}^{t_{2}} \overrightarrow{\boldsymbol{F}}_{i j} \mathrm{~d} t \tag{8.1} \label{8.1}\]
Impulse has \(\mathrm{N} \cdot \mathrm{s}\) as unit.
The momentum vector \(\overrightarrow{\boldsymbol{p}}_{i}\) is defined as the product of a mass and its velocity vector:
\[\overrightarrow{\boldsymbol{p}}_{i} \equiv m_{i} \overrightarrow{\boldsymbol{v}}_{i} \tag{8.2} \label{8.2}\]
The unit of momentum is \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\).
The impulse acting on a mass \(m_{i}\) equals its change in momentum.
\[\overrightarrow{\boldsymbol{p}}_{i}\left(t_{1}\right)+\overrightarrow{\boldsymbol{J}}_{i, 12}=\overrightarrow{\boldsymbol{p}}_{i}\left(t_{2}\right) \tag{8.3} \label{8.3}\]
Since this is a vector equation, it can be projected on the three coordinate axes to obtain three scalar equations along each of the axes. The principle of impulse and momentum is mainly useful to calculate velocity changes if the impulse is known.
The principle of impulse and momentum can be derived by integrating Newton’s second law over time:
\[\sum_{j} \int_{t_{1}}^{t_{2}} \overrightarrow{\boldsymbol{F}}_{i j} \mathrm{~d} t=\int_{t_{1}}^{t_{2}} m_{i} \overrightarrow{\boldsymbol{a}}_{i} \mathrm{~d} t \tag{8.4} \label{8.4}\]
The integral on the right can be evaluated to obtain:
\[\sum_{j} \int_{t_{1}}^{t_{2}} \overrightarrow{\boldsymbol{F}}_{i j} \mathrm{~d} t=\int_{t_{1}}^{t_{2}} m \frac{\mathrm{d} \overrightarrow{\boldsymbol{v}}_{i}}{\mathrm{~d} t} \mathrm{~d} t=m \overrightarrow{\boldsymbol{v}}_{i}\left(t_{2}\right)-m \overrightarrow{\boldsymbol{v}}_{i}\left(t_{1}\right) \tag{8.5} \label{8.5}\]
From Equation 8.1 and Equation 8.2 it follows that this equation is identical to Equation 8.3.