Relation between Kinetic Energy and Momentum

Relation between Kinetic Energy and Momentum Overview

The relation between kinetic energy and momentum lies in their dependence on velocity. Momentum is defined as the product of an object's mass and velocity, indicating that any object with mass "m" in motion possesses momentum. The amount of momentum an object has is determined by its mass and velocity. Hence, all objects have potential energy, which gets converted into kinetic energy when they start moving. For instance, a ball placed at the edge of a slope has potential energy, which gets converted into kinetic energy when it starts rolling down. Therefore, the mass of a moving object is referred to as "momentum," while the energy possessed by an object in motion is called "kinetic energy."

What is Kinetic Energy?

Kinetic energy refers to the energy acquired by an object due to its motion. It represents the amount of work needed to accelerate a specific mass from rest to a particular velocity. Once an object gains kinetic energy through acceleration, it retains that energy as long as its speed remains constant. When the object decelerates and comes to a stop, it will expend the same amount of work that was required to accelerate it. Additionally, any term in the Lagrangian of a system that has a time derivative is considered kinetic energy.

Kinetic Energy Formula

In mechanics, the kinetic energy of an object with mass 'm' that is moving at a speed 'v' is given by:

K.E. = ½ m. v2

Where,

m is the mass of the object, measured in kg.

v is the velocity of the object, measured in m/s.

This formula is a valid approximation in mechanics only when the speed 'v' is less than the speed of light.

What is Momentum?

In Newtonian physics, the vector quantity obtained by multiplying an object's mass with its velocity is known as linear momentum or translational momentum, simply referred to as momentum. It has both direction and magnitude and is measured in kilogram-meters per second (kgm/s) or newton-seconds in the International System of Units (SI).

Momentum Formula

The momentum of a moving object can be mathematically expressed as –

p = m.v

Where,

• p is the momentum.
• m is the mass of the object measured in kg.
• v is the velocity of an object measured in m/s.

Relation between Kinetic Energy and Momentum 

The momentum of an object is directly proportional to both its mass and velocity, meaning that an increase in either parameter will lead to a corresponding increase in momentum. Similarly, the kinetic energy of an object, which is the energy it possesses due to its motion, is also directly proportional to the mass and the square of the velocity of the object.

We know that,

K.E. = ½ m . v2

And, p = m. v

So we can write Kinetic Energy as:

K.E. = ½ (m . v) . v

Therefore,

K.E. = ½ p . v

Also, Kinetic Energy can be written as:

K.E. = ½ m. v2. (m / m)

è K.E. = ½ (m2. v2) / m

Therefore,

K.E. = p2 / 2m

Or we can also write:

p = √{2m(K.E.)}

Relation Between Kinetic Energy and Momentum: Calculations

To better understand the relation between kinetic energy and momentum, let's look at some examples of their calculations.

From the table, we can see that for the same object, increasing its velocity will increase both its kinetic energy and momentum. Additionally, objects with larger masses have large momenta but not necessarily larger kinetic energies.

Relation Between Kinetic Energy and Momentum: Application

Kinetic energy and momentum are important concepts in many areas of physics and engineering. For example:

1. In collision analysis, the conservation of momentum can be used to determine the velocities of objects before and after a collision.
2. In rocket propulsion, the momentum of the expelled fuel can be used to determine the thrust generated by the rocket.
3. In roller coasters, the kinetic energy of the coaster at the top of a hill is converted to potential energy as the coaster travels downhill, and the momentum of the coaster is used to propel it through loops and other elements.

Read more about the Efficiency Formula and Working Principle of Transformer.

Relation Between Kinetic Energy and Momentum: Things to Remember

1. The energy that results from an object's acceleration is referred to as kinetic energy, while momentum pertains to the mass of an object in motion.
2. The energy that an object acquires as it moves is its kinetic energy. This energy is the amount of work needed to accelerate a body with a specific mass from rest to a particular velocity.
3. Unless the object's velocity changes, the kinetic energy acquired during acceleration remains constant.
4. In Newtonian physics, the product of an object's mass and velocity is known as linear or translational momentum. This is a vector quantity with both magnitude and direction.
5. The momentum of an object is directly proportional to its mass and velocity. Hence, an object with a larger mass or velocity has a higher momentum.

Relation Between Kinetic Energy and Momentum: Sample Questions

Given a ball of mass 0.5 kg moving at a velocity of 10 m/s, calculate its momentum and kinetic energy.

Solution:

Momentum = mass x velocity

P = 0.5 kg x 10 m/s = 5 kg m/s

Kinetic Energy = 1/2 x mass x velocity^2

KE = 1/2 x 0.5 kg x (10 m/s)^2 = 25 J

If the momentum of a car is 2000 kg m/s and its mass is 500 kg, find its velocity and kinetic energy.

Solution:

Velocity = momentum / mass

v = 2000 kg m/s / 500 kg = 4 m/s

Kinetic Energy = 1/2 x mass x velocity^2

KE = 1/2 x 500 kg x (4 m/s)^2 = 4000 J

A bullet of mass 10 g is fired from a gun at a velocity of 400 m/s. Calculate its momentum and kinetic energy.

Solution:

Momentum = mass x velocity

P = 10 g x 400 m/s = 4 kg m/s

(Note: 1 g = 0.001 kg)

Kinetic Energy = 1/2 x mass x velocity^2

KE = 1/2 x 0.01 kg x (400 m/s)^2 = 800 J

If a block of mass 2 kg is moving with a velocity of 6 m/s, find its momentum and kinetic energy.

Solution:

Momentum = mass x velocity

P = 2 kg x 6 m/s = 12 kg m/s

Kinetic Energy = 1/2 x mass x velocity^2

KE = 1/2 x 2 kg x (6 m/s)^2 = 36 J

A train of mass 10000 kg is moving at a velocity of 20 m/s. Calculate its momentum and kinetic energy.

Solution:

Momentum = mass x velocity

P = 10000 kg x 20 m/s = 200000 kg m/s

Kinetic Energy = 1/2 x mass x velocity^2

KE = 1/2 x 10000 kg x (20 m/s)^2 = 2000000 J