Understand the Relation Between Kinetic Energy and Momentum

Momentum is simply the “mass in motion.” It means that every object of mass ‘m’ exhibiting a motion always possesses momentum. The amount of momentum that an object has relies on two variables: how much stuff is moving and how fast the stuff is moving.

So, basically, every object bears potential energy stored in itself. When this object sets into motion, it has some kinetic energy. For example, a rock sitting on the edge of a cliff. If the rock falls, the potential energy will be converted to kinetic energy, as the rock will be moving.

Now, a question arises: Is there any relationship between kinetic energy and momentum of an object? If so, is there any formula that describes the relationship between these two parameters?

This page will help you understand the Kinetic Energy and Momentum Relationship with the derivation of the formula of the same.

Formula for Kinetic Energy and Momentum

Since kinetic energy is directly proportional to half of the mass of an object and its velocity, it can be expressed as the following:

KE = \[\frac{1}{2}mv^{2}\]

Where 

m = mass of a body

V = velocity of an object due to change in its motion.

If the mass contains 1 kilogram and the velocity of a body is meters/second, the kinetic energy will be 1 kg per meter square and seconds square. The standard unit (S.I.) of kinetic energy gets measured in Joules.

Where 1 Joule = 1kg m2/s2

One can calculate momentum as mass multiplied with velocity. It is expressed in the following form.

P = m ∗ v

Where m = mass of a substance

v = velocity

Since, both the momentum and kinetic energy depend on velocity and mass, a change in one affects the other.

Kinetic Energy and Momentum

\[E=\frac{p^{2}}{2m}\]

Do you know what kinetic energy is? Well, kinetic energy is the energy that any substance has when it accelerates, whereas momentum is an object’s mass in motion. There is a kinetic energy and momentum relation due to their connection with mass and velocity.

From the above text, relation between kinetic energy and momentum can be mathematically shown as:

KE = \[\frac{1}{2}mv^{2}\]  and p = m * v

Here, KE = \[\frac{1}{2}m^{*}v^{*}v\]

Also, we see that \[m^{*}v\] = p

Therefore, KE = \[\frac{1}{2}p^{*}v\]

Alternative way to write the above equation is:

We know that KE =  \[\frac{1}{2}m^{*}v^{*}v\] 

Now, \[\frac{1(m^{*}v)^{2}}{2 m}\]

We know that p = mv

So, 

KE =\[\frac{1}{2}\frac{p^{2}}{m}\]  

Or

From the above formula(1) on kinetic energy and momentum relationship, we see that a body’s kinetic energy is equal to the product of momentum and half its velocity. Also, \[p^{2}=2mKE\]

Or

p =\[\sqrt{2mKE}\]  is the relation between linear momentum and kinetic energy.

Solved Questions on the Relationship between KE and p

Find some solved questions and answers on energy and momentum.

1. Momentum can change due to

1. Acceleration

2. Impulse

3. Force

4. All of these

Ans: d

2. A boxer quickly changes his move, bending his head backwards when someone tries to hit him on the head. What does this motion achieve?

1. Momentum increases, force decreases

2. Contact time increases as a result force decreases

3. Contact time decrease, increasing force

4. It helps to confuse the opponent

Ans: b

3. An object placed in a resting posting has which of the following properties?

1. Velocity

2. Potential energy

3. Kinetic energy

4. Momentum

Ans: b

4. The total energy in an object including rest energy in the world is:

1. Cannot change

2. Can decrease but not increase

3. May either decrease or increase

4. Can increase but not decrease

Ans: a

Though the study of energy and momentum is vast, we hope from the above discussion; you can understand the relation between kinetic energy and linear momentum. To access study material on related topics and concepts, install our Vedantu app and avail the online interactive sessions.