How to Find Momentum from Kinetic Energy: A Comprehensive Guide

Momentum and kinetic energy are two important concepts in physics. Momentum, represented by the letter ‘p’, is a property of moving objects that depends on their mass and velocity. On the other hand, kinetic energy, denoted by ‘KE’, is the energy possessed by an object due to its motion. In this blog post, we will explore how to find momentum from kinetic energy, how to find kinetic energy using momentum, and how to determine speed and mass using both momentum and kinetic energy.

How to Calculate Momentum from Kinetic Energy

The Mathematical Formula

The formula to calculate momentum is given by:

 p = \sqrt{2mKE}

Where:
– ‘p’ represents the momentum of the object
– ‘m’ is the mass of the object
– ‘KE’ stands for the kinetic energy of the object

Step-by-step Process

  1. Determine the mass of the object (in kilograms).
  2. Find the kinetic energy of the object (in joules).
  3. Substitute the values of mass and kinetic energy into the formula.
  4. Use a calculator to calculate the square root of the product of 2, mass, and kinetic energy.
  5. The resulting value will be the momentum of the object.

Worked-out Example

Momentum from Kinetic Energy 2

Let’s say we have an object with a mass of 2 kilograms and a kinetic energy of 50 joules. To find the momentum, we can use the formula mentioned earlier:

 p = \sqrt{2mKE}

Substituting the given values:

 p = \sqrt{2 \times 2 \times 50}

Calculating the square root:

 p = \sqrt{200}

This simplifies to:

 p \approx 14.14 \, \text{kg m/s}

Therefore, the momentum of the object is approximately 14.14 kg m/s.

How to Find Kinetic Energy using Momentum

The Mathematical Formula

To find kinetic energy using momentum, we can rearrange the formula mentioned earlier:

 KE = \frac{p^2}{2m}

Where:
– ‘KE’ represents the kinetic energy of the object
– ‘p’ is the momentum of the object
– ‘m’ stands for the mass of the object

Step-by-step Process

  1. Determine the momentum of the object (in kg m/s).
  2. Find the mass of the object (in kilograms).
  3. Substitute the values of momentum and mass into the formula.
  4. Use a calculator to calculate the square of the momentum, and then divide it by 2 times the mass.
  5. The resulting value will be the kinetic energy of the object.

Worked-out Example

Let’s consider an object with a momentum of 20 kg m/s and a mass of 5 kilograms. To find the kinetic energy, we can use the formula mentioned earlier:

 KE = \frac{p^2}{2m}

Substituting the given values:

 KE = \frac{20^2}{2 \times 5}

Calculating:

 KE = \frac{400}{10}

Simplifying:

 KE = 40 \, \text{joules}

Therefore, the kinetic energy of the object is 40 joules.

How to Find Speed and Mass using Momentum and Kinetic Energy

Finding Speed with Momentum and Kinetic Energy

The Mathematical Formula

To find the speed of an object using its momentum and kinetic energy, we can use the following formula:

 v = \sqrt{\frac{2KE}{m}}

Where:
– ‘v’ represents the speed of the object
– ‘KE’ is the kinetic energy of the object
– ‘m’ stands for the mass of the object

Step-by-step Process

  1. Determine the kinetic energy of the object (in joules).
  2. Find the mass of the object (in kilograms).
  3. Substitute the values of kinetic energy and mass into the formula.
  4. Use a calculator to calculate the product of 2 times the kinetic energy, and then divide it by the mass.
  5. Finally, take the square root of the result to find the speed of the object.

Worked-out Example

Momentum from Kinetic Energy 1

Suppose we have an object with a kinetic energy of 100 joules and a mass of 10 kilograms. To find the speed, we can use the formula mentioned earlier:

 v = \sqrt{\frac{2KE}{m}}

Substituting the given values:

 v = \sqrt{\frac{2 \times 100}{10}}

Simplifying:

 v = \sqrt{20}

Calculating the square root:

 v \approx 4.47 \, \text{m/s}

Therefore, the speed of the object is approximately 4.47 m/s.

Finding Mass given Momentum and Kinetic Energy

How to Find Momentum from Kinetic Energy
Image by Kraaiennest – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 4.0.

The Mathematical Formula

To find the mass of an object using its momentum and kinetic energy, we can rearrange the formula for momentum:

 m = \frac{p^2}{2KE}

Where:
– ‘m’ represents the mass of the object
– ‘p’ is the momentum of the object
– ‘KE’ stands for the kinetic energy of the object

Step-by-step Process

  1. Determine the momentum of the object (in kg m/s).
  2. Find the kinetic energy of the object (in joules).
  3. Substitute the values of momentum and kinetic energy into the formula.
  4. Use a calculator to calculate the square of the momentum, and then divide it by 2 times the kinetic energy.
  5. The resulting value will be the mass of the object.

Worked-out Example

Momentum from Kinetic Energy 3

Let’s assume an object has a momentum of 30 kg m/s and a kinetic energy of 150 joules. To find the mass, we can use the formula mentioned earlier:

 m = \frac{p^2}{2KE}

Substituting the given values:

 m = \frac{30^2}{2 \times 150}

Calculating:

 m = \frac{900}{300}

Simplifying:

 m = 3 \, \text{kg}

Therefore, the mass of the object is 3 kilograms.

In this blog post, we explored how to find momentum from kinetic energy, how to find kinetic energy using momentum, and how to determine speed and mass using both momentum and kinetic energy. By understanding these concepts and using the appropriate formulas, we can gain insights into the dynamics of moving objects and their energy interactions. Keep practicing these calculations to strengthen your understanding of physics and its applications.

Numerical Problems on How to Find Momentum from Kinetic Energy

Problem 1:

A particle of mass 2 kg is moving with a kinetic energy of 50 J. Calculate its momentum.

Solution:

Given:
Mass of the particle, m = 2 kg
Kinetic energy, K = 50 J

We know that the relation between kinetic energy and momentum is given by:

 K = \frac{1}{2} mv^2

where m is the mass of the particle and v is its velocity.

Rearranging the equation, we can find the velocity of the particle:

 v = \sqrt{\frac{2K}{m}} = \sqrt{\frac{2 \times 50}{2}} = 5 \, \text{m/s}

The momentum of the particle can be calculated using the formula:

 p = mv = 2 \times 5 = 10 \, \text{kg.m/s}

Therefore, the momentum of the particle is 10 kg.m/s.

Problem 2:

An object with a mass of 0.5 kg has a kinetic energy of 25 J. Find its momentum.

Solution:

Given:
Mass of the object, m = 0.5 kg
Kinetic energy, K = 25 J

Using the relation between kinetic energy and momentum, we have:

 K = \frac{1}{2} mv^2

Solving for velocity, we get:

 v = \sqrt{\frac{2K}{m}} = \sqrt{\frac{2 \times 25}{0.5}} = 10 \, \text{m/s}

The momentum of the object can be calculated as:

 p = mv = 0.5 \times 10 = 5 \, \text{kg.m/s}

Therefore, the momentum of the object is 5 kg.m/s.

Problem 3:

A car has a kinetic energy of 4000 J and its mass is 1000 kg. Determine its momentum.

Solution:

Given:
Mass of the car, m = 1000 kg
Kinetic energy, K = 4000 J

From the equation relating kinetic energy and momentum, we have:

 K = \frac{1}{2} mv^2

Rearranging the equation to solve for velocity, we get:

 v = \sqrt{\frac{2K}{m}} = \sqrt{\frac{2 \times 4000}{1000}} = 8 \, \text{m/s}

The momentum of the car can be calculated using the formula:

 p = mv = 1000 \times 8 = 8000 \, \text{kg.m/s}

Therefore, the momentum of the car is 8000 kg.m/s.

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