How to Derive the Equation Between Kinetic Energy and Momentum

It is easy to see the connection between kinetic energy and forward velocity.  Kine, kinetic or momentum (motion), is the energy a body has. As a result, kinetic energy and momentum have a connection. A more in-depth discussion of the relationship will take place later on in the article. It’s impossible to grasp the relation between kinetic energy and momentum without knowing how they’re generated. As a result, we must first understand kinetic energy and momentum to establish the relationship between the two. Starting with kinetic energy, we’ll go from there.

What is kinetic energy?

Kinetic energy refers to the energy a body has due to the motion of its constituent components. Kinetic energy, on the other hand, is a result of particle motion. 

More kinetic energy is released while moving faster, and less kinetic energy is released when moving slower. We spoke about kinetic energy in the last section since it is based on particle movement or motion. However, momentum is now used to describe kinetic energy. 

  • The unit of kinetic energy is the letter K.E.
  • It’s a a scalar quantity.
  • There is no direction to kinetic energy.

Because of kinetic energy, a hammer that is in motion will force a nail into the wall. Similar to how the kinetic energy of a moving bullet may pierce a steel plate. In reality, kinetic energy is present in everything around us that moves. To put it another way, everything in our environment has kinetic energy.

What Do You Mean by Momentum?

Mass multiplied by velocity equals momentum. That is to say; momentum is proportional to mass and velocity. That is to say, when a body’s mass and velocity rise, momentum increases as well.

  • The symbol for momentum is P.
  • It is a vector quantity.
  • The momentum equation is written as P = mv.

No matter how fast an item moves, its momentum is always equal to the product of its mass and velocity. To put it another way, if the moving body’s velocity rises, then its momentum increases as well.

Relationship between Kinetic Energy and Momentum

To put it another way, kinetic energy and momentum are inversely proportional. As a result, large objects have more kinetic energy and momentum than tiny objects moving slowly while travelling at high speeds. This means that kinetic energy is proportional to the mass and velocity of the object that generates it. 

As a result, as the mass of a body double, so does its kinetic energy and momentum. The kinetic energy doubles when the velocity is doubled, not when the mass is doubled. As long as you’re not talking about momentum. To put it another way, kinetic energy has the same mathematical relationship to the momentum that work does. 

K.E = 1/2 mv2

K.E. equals half of a Pv

If you multiply by two, you get the following answer:

This relationship states that as kinetic energy rises, so does momentum. However, momentum decreases if kinetic energy decreases. We recently learned that the formula Kinetic Energy = 1/2 mv2 gives us the kinetic energy of a mass ‘m’ travelling at a velocity ‘v.’ 

To summarise, kinetic energy is directly proportional to mass and may be calculated using the formula below: The kinetic energy of a body is related to its velocity squared. Momentum = PV is a formula that may be used to calculate the mass and velocity of a moving body. Momentum is a function of mass and may be expressed as a ratio. 

The body’s velocity is inversely related to its momentum. As a result, there are several parallels between kinetic energy and momentum. Both have a mass and a velocity that are inversely proportional.

Derivation of equation between Kinetic Energy to Momentum

Consider a body with mass ‘m,’ is travelling at a velocity of ‘v,’ through a medium. After a while, its speed slows down until it comes to a halt. The body’s final velocity is 0 meters per second. We’ll use’s’ as our unit of total distance travelled for the whole excursion. The body has worked while travelling long distances.

It’s calculated as follows: Work completed = Force Distance W is equal to F minus s.

The third law of motion says that if a body moves with an initial velocity “v,” a final velocity “V,” acceleration “a,” and covers a distance “s,” it moves as follows:

 

(2) = (4) (2)as ——————— (1)

The following is an illustration of what we’re talking about:

Body’s initial speed is given by the equation v. (supposed) Body’s final speed = V = 0 ( the body stops) Body speeding up = -a (due to retardation) Simplifying, the distance travelled is equal to s. (let)

Equation (1) now gives us: (0)2 = v2 – 2as v2 = 2as. ——————- (2)

The following is derived from Newton’s second law of motion:

The formula is as follows: F = ma a = F/m

Now that we have the value of ‘a’ in equation (2), we obtain the following value:

To recap, kinetic energy is the amount of effort that the body does while moving.

In other words, the energy is equal to half of the work done in a second. kinetic energy is equal to half of the mv2 ——————— (3)

A body’s momentum is equal to the product of its mass and velocity while travelling at a velocity.

This means that Momentum = Mass x Velocity P = mv Equation (3): Kinetic energy = 1/2 mv2

In other words, K.E is equal to half of the mass divided by the volume. If K.E is equal to half the square of the second dimension, then,  K.E = p2/2m

Since K.E = p2/2m, we may write it as: K.E = 2p2/m

Conclusion

Hence, we may conclude from this deduction that kinetic energy and momentum have a direct relationship. The kinetic energy of a body is inversely related to its mass and proportional to the square of its momentum.

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