What is Linear Momentum?

Linear Momentum is the product of an object’s mass and its velocity. It is a vector quantity, meaning it has both magnitude and direction.

Linear momentum, generally known as the momentum of a body, is defined as the total quantity of motion possessed by the moving body, and is measured as the product of the mass of the particle and its velocity.

Linear momentum $\overrightarrow{p}$ is a vector quantity, whose direction is the same as the velocity of the body. The S.I. unit of momentum is given by $kgm{s}^{-1}$. If a body of mass m is moving with a velocity $\overrightarrow{v}$, then its momentum is $$\overrightarrow{p}=m\overrightarrow{v}$$.

Newton’s Second Law of Motion: Momentum

According to Newton’s second law of motion and momentum, when the same force acts on two bodies of different masses for the same interval of time, we can observe different effects on the objects. The lighter object moves with a higher velocity than the heavier object. However, the change in momentum of both bodies is the same.

“The rate of change of momentum of a body is directly proportional to the applied force, and the direction of the change is in the same direction as the force.”

From Newton’s Second Law of Motion, for a fixed mass particle: F = ma

\begin{array}{l} \frac{d\overrightarrow{F}}{dt}=\frac{d}{dt}\left( m\overrightarrow{a} \right)=m\frac{d^2\overrightarrow{v}}{dt^2}=\frac{d^2}{dt^2}\left( m\overrightarrow{v} \right)=\frac{d\overrightarrow{p}}{dt} \end{array}

For a system of n particles with masses $m_1,m_2,m_3,\dots ,m_n$ and velocities $$\vec{v_1}, \vec{v_2}, \vec{v_3}, \ldots, \vec{v_n}$$ respectively, then the net momentum of the system is $$\sum_{i=1}^{n} m_i \vec{v_i}$$.

$$(\overrightarrow{{{p}_{net}}} = \overrightarrow{{{p}_{1}}}+\overrightarrow{{{p}_{2}}}+\overrightarrow{{{p}_{3}}}+…+\overrightarrow{{{p}_{n}}} = {{m}_{1}}\overrightarrow{{{v}_{1}}}+{{m}_{2}}\overrightarrow{{{v}_{2}}}+{{m}_{3}}\overrightarrow{{{v}_{3}}}+…+{{m}_{n}}\overrightarrow{{{v}_{n}}})$$

\begin{array}{l} \overrightarrow{p_{net}} = M \overrightarrow{V_{cm}} \end{array}

Differentiating the above expression with respect to time

\begin{array}{l}\frac{d\overrightarrow{{p_{net}}}}{dt} = M\frac{d\overrightarrow{{V_{cm}}}}{dt}\end{array}

$$\overrightarrow{F_{net}} = M\overrightarrow{a_{cm}}$$

And also

\begin{array}{l}\frac{d\overrightarrow{{{p}_{net}}}}{dt}=\overrightarrow{{{F}_{net}}}\end{array}

The kinetic energy of a system may be expressed in terms of its linear momentum as well.

\begin{array}{l}m=\frac{p}{v}\end{array}

$$\begin{array}{l}{p}^{2} = {m}^{2}{v}^{2} = 2m\left(\frac{1}{2}m{v}^{2}\right) = 2mK\end{array}$$

Law of Conservation of Momentum

The law of conservation of momentum states that if the net force acting on a body is equal to zero, then the momentum of the body remains constant.

\(\begin{array}{l}{F_{net}} = 0\end{array}\)

$$(\frac{d{{p}_{net}}}{dt} = 0)$$

Therefore, pnet = 0 or pnet = constant

If the velocity of centre of mass is equal to zero, (v_cm = 0), then from

$\overrightarrow{p_{net}}=M\overrightarrow{V_{cm}}$

we get,

$$\overrightarrow{{{p}_{net}}}=0.$$

If the velocity of the center of mass is constant (v_cm = constant) then we get $$\overrightarrow{p_{net}}= \text{constant.}$$

This is known as the Law of Conservation of Linear Momentum of a System of Particles.

Frequently Asked Questions on Momentum

Define the term linear momentum of a body.

Linear momentum of a body is the product of its mass and velocity, and is denoted by the symbol $p$ (momentum). It is a vector quantity, meaning that it has both magnitude and direction.

The linear momentum of a body is the amount of motion it contains, and is defined as the product of mass and velocity.

The rate of change of momentum of the body is 2 kgm/s. When is the force acting on the body?

The force acting on the body is equal to the rate of change of momentum of the body, which is 2 kgm/s.

Force = Rate of change of Momentum

Therefore, Force = 2 N

The total momentum of the universe remains constant. Is this statement true?

Yes, this statement is true.

The law of conservation of momentum is a general law applicable to all isolated systems, making the given statement true.

Linear momentum is a vector or scalar quantity?

Linear momentum is a vector quantity.