Derivation of Kinetic Energy Formula

The word Kinetic energy came from the French word travail mécanique (mechanical work) or quantité de travail (quantity of work). The kinetic energy of a body is the energy possessed by the body by virtue of its motion.

For example, there are various applications of kinetic energy:

• A nail is driven into a wooden block on the amount of kinetic energy of the hammer striking the nail.

• A baseball thrown by a pitcher has a small mass but a large amount of kinetic energy due to high velocity.
  
  

On this page, we will learn about the following:

• The formula for kinetic energy

• Derive the formula of kinetic energy

• Kinetic energy equation derivation

• Kinetic energy derivation calculus

• Derivation of kinetic energy using algebra

The Formula for Kinetic Energy

A dancing man is said to be more energetic than a snoring man. In physics, a moving particle is said to have more energy than the particle at rest. Quantitatively the energy of the moving particle is defined by:

This is the kinetic energy of the particle.

Therefore, the sum of the kinetic energy of a system of particles is the sum of all its constituent particles. Which is given by:

The kinetic energy of a particle or a system of particles can increase or decrease or remain constant as time passes.

Deriving The Kinetic Energy Formula by Algebra

The kinetic energy can be obtained by either of the following:

• The amount of work done in stopping any moving object.

• The amount of work done in giving the velocity to the body from the state of rest.

Let’s consider the second case:

Suppose m =  mass of the body at rest.

u = Initial velocity of the body.

                F = The force applied to the body

                a = acceleration produced in the body in the direction of force applied

               v = The velocity acquired by the body while moving a distance s.

K∆S = W = m∆a….(1)

Using the third equation of motion:

v²-u²=2as

a=\[\frac{v^2-u^2}{2s}\]….. (2)

Where,

v- The final velocity of the body

u- The initial velocity of the body

S- The total displacement of the body

We know that according to Newton’s law of motion:

F=ma……(3)

Substituting the value of ‘a’ in eq(3) we get:

\[F=m(\frac{v^2-u^2}{2s})\]……(4)

Combining eq(1) and (4), we get that:

\[\Delta K=m(\frac{v^2-u^2}{2s})\]……..(5)

Substituting u = 0  in eq(5), we get that:

Work done on the body is given by:

W = Force x distance….(7)

This statement states that a work W is done by a body to move from one position to another by a distance s when the force F is applied to a body at rest.

Putting u = 0  in eq(2), we get that:

\[a=(\frac{v^2}{2s})\]

F = m  a

Therefore, we get:

\[F=m(\frac{v^2}{2s})\]

Putting the value of F in eq(7), we get that

\[W=m(\frac{v^2}{2s}\times S=\frac{1}{2}mv^2)\]

This work done on the body is because of the Kinetic energy (K.E) of the body.

Derivation of Kinetic Energy Formula by Calculus

The formula for kinetic energy can be obtained by the method of calculus:

Suppose

m = mass of a body

u =  Initial velocity of the body 

\[\vec{F}\]  =  The force applied to the body in the direction of the motion

\[\vec{ds}\] = A small displacement of the body in the direction of motion

The small amount of work done by the force will be:

dW =  \[\vec{F}\]  . \[\vec{ds}\] = F ds Cos 0°  = Fds  (∵  Cos 0° = 1)

dW = F ds = m  \[\vec{a}\] ds    (∵  \[\vec{F}\] = m   \[\vec{a}\]  )  ( \[\vec{a}\]  is an acceleration produced by the force)

\[dW=m(\frac{d\vec{v}}{dt})ds\] (∵\[\vec{a}=\frac{d\vec{v}}{dt}ds\]

\[dW=m(\frac{ds}{dt})dv\]

\[dW=mvdv\] (∵\[\vec{v}=\frac{ds}{dt}\])

Total work done by the force is increasing the velocity of the body from  u  to v is:

\[W=\int_{u}^{v} mvdv=mv[\frac{(1+1)}{(1+1)}]^{v}_{u}=[\frac{mv^2}{2}]^{v}_{u}\]

\Rightarrow \[\frac{1}{2}m(u^2-v^2)\]

Total work done by the force is increasing the velocity of the body from  u  to v is:

\[\int_{0}^{W}dW=\int_{u}^{v}mvdv\]

\[\Rightarrow W=m[\frac{v^2}{2}]_{u}^{v}=\frac{m}{2}(u^2-v^2)\]

\[\Rightarrow W=\frac{mv^2}{2}-\frac{mu^2}{2}\]=Final K.E - Initial K.E…..(8)

The work done by a force is a measure of the change in kinetic energy of the body which proves the work-energy principle.

On putting u = 0  in eq (8) , we get that:

\[W=\frac{1}{2}mv^2\]-0

Importance Of Kinetic energy:

Kinetic energy is the energy that an object receives from its movement.

If we wish to accelerate an object then we need to apply force. In order to apply force, we need to work. Once the work is completed the energy is transmitted to the subject which means that the item will then be moving at a constant speed. The energy transfer is known as kinetic energy and is dependent on the mass and the speed that is achieved.

The energy of kinetics can transfer between different objects and later be transformed into different forms of energy. For instance, a flying squirrel could encounter chipmunks that are stationary. After the collision, a portion of the initial energy of the squirrel may be transferred to the chipmunk's body or transformed into an alternative type of energy.

1) It's a key component in the calculation of work.

For example, if you lift an object with a certain amount of force, that object has gained kinetic energy. The more distance the object travels as it's being lifted, the more kinetic energy it will have. Once you've moved the object to its final destination, you then must use another type of energy (potential energy) to keep it there. This is why lifting something heavy can be so tiring - you're not just using muscle power to do the work, you're also expending energy to overcome gravity and keep the object in place.

2) It's frequently used in the study of motion.

For example, engineers need to know how much kinetic energy a moving object will have in order to design things like cars and airplanes that are safe for people to use. The faster an object is traveling, the more kinetic energy it has - and if that object collides with something else, that energy can be converted into other forms (such as heat or sound). This is why it's important for car designers to make sure that their vehicles can withstand crashes.

3) It can be harnessed to do work.

For example, you can use kinetic energy to power a machine by converting it into electrical energy. This is how wind turbines and hydroelectric dams work - they capture the kinetic energy of flowing water or air and turn it into electricity that can be used to power homes and businesses.

4) It's a fundamental part of Einstein's famous formula E = mc².

Although this equation is most often used in reference to nuclear energy, remember that all matter has both mass and kinetic energy - the faster an object moves, the greater it’s kinetic energy will be. When you add up all forms of potential and kinetic energy possessed by a body, you get its total mass (m). If something contains more potential or kinetic energy than it does mass, then that extra amount can turn into other types of energies such as heat or light. This holds true whether we're talking about atoms or entire galaxies!

Summary

• The force does some work on a body, the kinetic energy increases by the same amount. Thus, according to this principle, work and energy are equivalent to each other.

• A force is necessary to change the kinetic energy of the particle. If the resultant force acting on a particle is perpendicular to its velocity, the speed does not change and hence the kinetic energy does not change.

Example:

Q1: Calculate the average frictional force needed to stop a car weighing 600 kg at a distance of 35 m, if the initial speed is 36 km/h.

Solution:  Given m = 600 kg, s =35 m

                       v =  final velocity = 0  (Since the body will stop finally)

                        u= 36 km/h       

                u = 36 kmph = 10 m/s

According to work-energy principle,

W = Change in K.E. =\[\frac{1}{2}m(v^2-u^2)\]

F x s =\[\frac{1}{2}m(v^2-u^2)\]

and, W = F x s 

F x 35 = \[\frac{1}{2}600\times (0-10^2)\]

On solving, we get that: