Section 5.4 Impulse and Momentum
The quantity of motion contained in a body is known as momentum. It is defined as the product of mass and velocity of a body. It is a vector quantity.
\begin{equation*}
\vec{p}=m\vec{v}
\end{equation*}
Its unit is \(kg m/s^{2}\) in SI system. The momentum has same direction as velocity of the object. There are two types of momentum, linear momentum and angular momentum. The object moving with a linear velocity, \(v\) has linear momentum and the object spinning with angular velocity, \(\omega\) possesses angular momentum. Here only linear momentum (normally called momentum) is being discussed. The momentum of a system of particles is the sum of the momenta of the individual particles.
\begin{equation*}
\vec{p}_{tot}=\sum_{i} m_{i}\vec{v}_{i}
\end{equation*}
From Newton’s II law of motion, force acting on a particle is the rate of change of momentum and its direction is along that change
\begin{equation*}
\vec{F}_{tot}=\frac{\,d\vec{p}}{\,dt}
\end{equation*}
For a system of particles total force can be,
\begin{equation*}
\vec{F}_{tot}=\sum_{i}\vec{F}_{i}=\sum_{i}\frac{\,d\vec{p}_{i}}{\,dt} = \frac{\,d\vec{p}_{tot}}{\,dt}
\end{equation*}
In this expression
\begin{equation*}
\sum_{i}\vec{F}_{i}=\sum_{i}\vec{F}^{ext}_{i}
\end{equation*}
because the internal forces are Newton’s III law pairs and they are directed along the lines between particles of the system and hence cancel each other out. Therefore, the total force acting on the system is
\begin{equation*}
\vec{F}_{tot} = \sum_{i}\vec{F}^{ext}_{i} =\frac{\,d\vec{p}_{tot}}{\,dt}
\end{equation*}
\begin{equation*}
\text{or,}\quad \vec{p}_{tot} =\int\,d\vec{p}_{tot}= \int\vec{F}_{tot}\,dt
\end{equation*}
A force acting on an object for a certain time changes the momentum of that object, such change in momentum is called Impulse. If total external force acting on a system is zero, then total linear momentum of the system,
\begin{equation*}
\vec{p}_{tot} = constant.
\end{equation*}
This is known as the principle of conservation of momentum. That is
\begin{equation*}
\vec{p}_{tot} =\vec{p}_{i}=\vec{p}_{f}
\end{equation*}
where \(\vec{p}_{i}\) and \(\vec{p}_{f}\) are the initial and final momenta of a system. Momentum conservation is very useful in collision problems because the collision force is internal and hence does not change the total momentum.
The impulse of a collision is the total momentum transferred during the collision, where a collision is an event where a very large force is exerted over a very short time interval \(\Delta t\text{.}\) During that interval magnitude of force is continuously changing, Therefore, the impulse of a force is the product of the average force and the time interval during which the force acts. That is,
\begin{equation*}
\vec{J}=\vec{F}_{av}\,dt=\,d\vec{p}
\end{equation*}
where,
\begin{equation*}
\vec{F}_{av} = \frac{1}{\Delta t}\int\limits_{0}^{t}\vec{F}\,dt
\end{equation*}
Impulse is a vector quantity and has the same direction as the average force. Unit of impulse is Ns.
