Linear Momentum – Definition, Formula, Examples

Linear Momentum is the physical property of a mass moving with a certain velocity in a straight line path. Linear momentum is given as the product of the mass of the object and the velocity with which it is moving. In the physical world, we can realize the significance of linear momentum from the fact that a heavier body with a higher velocity creates more destruction than a smaller body with the same velocity.

In this article, we will understand the meaning of linear momentum, the formula of linear momentum, linear momentum of a system of particles, and the law of conversion of linear momentum.

What is Linear Momentum?

Linear momentum is a fundamental concept in physics that describes the motion of an object in a straight line. It is a vector quantity defined as the product of an object’s mass and its velocity. It means a body with a larger mass and a faster speed has more momentum than one with a smaller mass and a slower speed. We can understand the importance of momentum with the fact that the object with more momentum will cause more destruction than the one with smaller momentum.

Linear Momentum Formula

Linear momentum (p) is a vector quantity, which is the product of an object’s mass (m) and its velocity (v). Mathematically, it is represented as:

p = m × v

Where,

• p = linear momentum
• m = mass of the object
• v = velocity of the object.

Linear Momentum of a System of Particles

Linear momentum of a system of particles arises from interactions among the particles, leading to the principle of conservation of linear momentum (P). This principle states that the total linear momentum of the system, equal to the sum of the linear momenta of every individual particle, remains constant. If a system comprises n particles, the total linear momentum is given by the sum of the linear momenta of each particle.

Where:

• P = total linear momentum of the system
• m_i = mass of the i^th particle
• v_i = velocity of the i^th particle

For Example:

The linear momentum of the particle is:

P = mv

For ‘n’ no. of particles total linear momentum is,

P = P₁ + P₂ + P₃ +….+ P_n

Each of momentum is written as

m₁v₁ + m₂v₂ +………..+ m_nv_n

We know that the velocity of the center of mass is:

MV = ∑m_iv_i ,

So comparing these equations we get,

P = M V

Where M is the center of mass, and V is the velocity of its center of mass.

Law of Conservation of Linear Momentum

The law conservation of linear momentum is a fundamental law in physics that states that the total linear momentum of a closed system remains constant if no external forces act upon it. In other words, for a closed system, total momentum before an event, such as a collision or explosion is equal to the total momentum after the occurrence of the event. Mathematically, this principle can be expressed as:

P_initial = P_final

Let’s say a there are two bodies with mass m₁ and m₂. These bodies are moving in a straight with velocity u₁ and u₂ facing each other. After some time they collide with each other such that their new velocities are v₁ and v₂. Then according to law of conservation of momentum, the total momentum before and after collision will remain same. We can express mathematically it as

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

where,

• m₁u₁ + m₂u₂ is the total momentum of two bodies before collision
• m₁v₁ + m₂v₂ is the total momentum of two bodies after collision

The law of conservation of linear momentum can be understood as follows:

Since P = mv

Differentiating the above equation we get,

(d/dt)P = md/dt(v)

(d/dt)P = ma,

From Newton′s second law of motion, F=ma

Hence, (d/dt)P = F

Thus, acting force is the rate of change of the momentum of the object.

Thus, if the total external force acting on the system is zero, then

(d/dt)P = 0

Solved Examples on Linear Momentum

Example 1: Find the linear momentum for a body whose mass is 10 kg and has a speed of 20m/s.

Solution:

Given parameters are,

m = 10 kg

v = 20m/s

Linear momentum formula is expressed as,

P = mv

p = 10 × 20

p = 200 kg·m/s

Example 2: The linear momentum of an object is 40 kg m/s and the mass of the object is 5kg..000 Find out the speed of the moving object.

Solution:

Given parameters are,

m = 5 kg

P = 40 kgm/s

Linear momentum formula is given by,

P = mv

v = P / m

v = 40 / 5

v = 8 m/s

Example 3: A have tennis ball 0.06kg is served with a velocity 30m/s calculate its linear momentum.

Solution:

p = m⋅v

p = 0.06 kg × 30 m/s = 1.8 kgm/s

Example 4: A bullet of 0.01kg mass is fired from a gun with a velocity of 400m/s, find out its kinetic momentum.

Solution:

p = m⋅v

p = 0.01 kg × 400 m/s

= 4 kgm/s

Example 5: A hockey play with mass 0.16kg is slapped with force 50N for 0.2 sec. Find its remaining linear momentum after the collision.

Solution:

Impulse = F·t

Impulse = 50 N × 0.2s = 10Ns

Change in momentum = Impulse

Change in momentum = 10Ns

Linear Momentum Practice Questions

Q1. A car with a mass of 1200 kg is moving at a velocity of 25 m/s. Calculate its linear momentum.

Q2. An object has a linear momentum of 150 kg m/s. If its mass is 10 kg, what is its velocity?

Q3. Two objects, one with a mass of 5 kg and the other with a mass of 8 kg, are moving with velocities of 10 m/s and 6 m/s respectively in the same direction. Calculate the total linear momentum of the system.

Q4. A tennis ball of mass 0.1 kg is served with a velocity of 30 m/s. Determine its linear momentum.

Q5. A truck has a linear momentum of 4800 kg m/s. If its velocity is 40 m/s, what is its mass?

Q6. A bullet of mass 0.02 kg is fired from a gun with a velocity of 500 m/s. Find its linear momentum.

Q7. A soccer player kicks a ball with a force of 100 N for 0.1 seconds. If the mass of the ball is 0.5 kg, what is its change in momentum?

Linear Momentum FAQs

What is meaning term linear momentum?

Linear Momentum is the physical quantity that describes the motion of an object moving in a straight line

Is linear momentum conserved in a collision?

Yes, linear momentum is conserved in a collision.

What is the relationship between the mass and the velocity and how is linear momentum associated with it?

There is a linear connection between mass (m) and the speed (v) for the momentum (p). It would be obtained with the help of the formula p = m⋅v

When momentum is called linear?

Momentum is called linear when it refers to the motion of an object along a straight line.

What is linear momentum and SI unit?

Linear momentum is the product of an object’s mass and its velocity. Its SI unit is kilogram meter per second (kg·m/s)

What is linear and angular momentum?

Linear momentum describes the motion of an object along a straight line, while angular momentum relates to the rotational motion of an object around an axis.

What is rate of linear momentum called?

The rate of change of linear momentum is called force, as described by Newton’s second law of motion.