Principle of Conversation of Linear Motion

Linear momentum

Momentum is the motion contained in a body. The quantity of motion possessed by a body depends on upon both of its mass and velocity. So, the product of mass and velocity is the measure of the momentum,

$$\vec p = m\vec v $$

It is also called linear momentum. It is a vector quantity whose direction is in the direction of the velocity. Its unit is kg m/s in SI-units and dimension, [ML^-1].

Principle of Conversation of Linear Motion

The law of conservation of linear momentum states that if no external forces act on the system of two colliding objects, then the vector sum of linear momentum of each body remains constant and is not affected by their mutual interaction.

Let us consider an isolated system of n particles having initial momentum p₁, p₂ …..p_n. Due to the collision, let the momentum of the particles after collision be p₁’, p₂’ ….. p_n’ respectively. Then according to the principle of conservation of linear momentum, in the absence of external force,

$$p_1 + p_2 + \dots p_n = p_1 ‘ + p_2 ‘ + \dots p_n ‘ $$

For verification, we consider a collision between two spheres A and B having masses of m₁ and m₂ respectively. Let u₁ and u₂ be the velocities of the spheres before collision such that u₁ > u₂ and moving on the same straight line as shown in the figure. After collision, let their velocities be v₁ and v₂ on the same line. If they collide each other for short interval of time t, each sphere exerts a force on the other sphere and so, the force experienced by A is given as

\begin{align*} F_2 &= \frac {\text {change in momentum}}{\text {time}} = \frac {m_1v_1 – m_1 u_1}{t} \\ \text {Similarly, force experienced by B is } \\ F_1 &= \frac {\text {change in momentum}}{\text {time}} = \frac {m_2v_2 – m_2 u_2}{t}\\ \end{align*}

According to Newton’s Third law of motion, the forced experienced by A and B are equal and opposite \begin{align*} \\ F_1 &= -F_2 \\ \text {or,} \: \frac {m_1 (v_1 –u_1)} {t} &= -\frac {m_2 (v_2 – u_2) } {t} \\ \text {or,} \: m_1v_1 –m_1v_1 &= -m_2v_2 + m_2u_2 \\ \text {or,} \: m_1u_1 + m_2u_2 &= m_1v_1 + m_2v_2 \\ \end{align*}

This proves that total momentum before collision is equal to the total momentum after collision if no external forces at on them prove the principle of conservation of linear momentum.