Derivation of Kinetic Energy - Definition, Equation, Example, FAQs

Key points of this chapter

• Derivation of kinetic energy formula.

Define kinetic energy and derive an expression for it

Kinetic energy:- The kinetic energy of a moving body is measured by the amount of work that has been done in bringing the body from rest position to its present position, or which the body can do in moving from its present situation to the rest position.

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Let a body mass m be in the rest position. When we implement a constant force F on the body, it starts moving under an acceleration. Let a be the acceleration, according to Newton’s second law, we have

a = F/m

Suppose the body obtains a velocity v in moving a distance s. According to the reaction

v² = 2as = 2 × (F/m) × s

F × s = 1/2mv²

But F × s is a work W which the force F has done on the body in moving at a distance s. It is due to this work that the body has self-acquired the capacity of doing work. This is the measure of the kinetic energy of the body. Hence, If we represent the Kinetic energy of a body by K then,

K = W = 1/2mv²

This is the expression for kinetic energy.

Kinetic energy =1/2 × mass × (velocity)²

Thus, the kinetic energy of a moving body is equal to half the product of the mass (m) of the body and the square of its speed v². In this formula, v occurs in the second power and so the speed has a larger effect, compared to the mass, on the kinetic energy. It is because of this reason that the bullet fired from a gun injures seriously inspite of its very small mass.

The unit of kinetic energy is Joule and the dimensional formula of the kinetic energy of [ML^²T^-²].

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Derive the expression of kinetic energy:- We can derive the formula of kinetic energy or derive the formula of kinetic energy class 9 or derive an expression for the kinetic energy of an object by the given method.

Although the above result has been obtained for a constant force acting on the body, it also holds if the force is variable in magnitude or in direction, or in both.

Let a force vector F of variable magnitude act on an object (body) of mass m which is initially at rest. The work done by the force in an infinitely small displacement vector ds of the body in the direction of the force is

dW = F.ds = Fdscos0 = Fds ….(i)

If a be the acceleration produced in the body during the displacement ds, then by Newton’s second law, we have

F= ma = mdv/dt [∴a = dv/dt]

Substituting this value of F in eqn. (i), we have

dW = (mdv/dt)ds

= m(ds/dt) dv = m v dv [∴ v = ds/dt]

Therefore, the total work done by the force in increasing the velocity of the body from 0 to v is

This is the kinetic energy of the body.

K = W = 1/2mv²

Kinetic energy is a scalar quantity.

So, this is the kinetic energy formula derivation and mathematical expression of kinetic energy.

Derive an expression for kinetic energy class 9: Let a body of mass m, moving with a velocity v, be acted upon by a retarding force F so that the body comes to rest over a distance s.

Here, the initial velocity is v and the final velocity is zero and it travels distances s.

Applying v² = u² + 2as

0 = v² + 2as (here a is negative)

Or v = 2as, ( u = v, v = 0)

s = v²/2a

Now, KE = work done by the force F in bringing the body to rest

= F× s

= (m × a ) × s

= (m × a ) × v²/2a

KE = 1/2mv²

This is the mathematical expression for kinetic energy

So, we can derive the equation of kinetic energy.

The KE is directly proportional to the mass of the body and inversely proportional to the square of the velocity of the body.

Work is done with the help of kinetic energy.

Work energy theorem:-

According to this theorem, the work done by a force in displacing a body is equal to the variation in the kinetic energy of the body. Suppose, a body of mass m is running with an initial velocity of u. When a force F works upon it, its velocity rises from u to v. Then, the work done by the force is given by

= 1/2 mv²- 1/2mu²

= Final velocity - initial velocity

= change in kinetic energy = ΔK (say)

Thus, W = ΔK

This is the mathematical declaration of the work-energy theorem.

In the above, we discuss the kinetic energy ( derive expression for kinetic energy ) in classical mechanics. Now we discuss the relativistic kinetic energy ( the kinetic energy of a particle ).

The kinetic energy of a body depends upon the reference of the frame.

The relativistic kinetic energy formula

KE = mc² - m₀c²

K = (m - m₀)c²

{If the velocity of the moving particle is nearly equal to the velocity of the light then the mass of the particle will not remain constant. It will change according to the relation

m = m₀/√( 1 - v²/c²)

Where m₀ is the rest mass of the particle.}

Kinetic energy types:-

Thermal energy, radiant energy sound energy, mechanical energy, and electrical energy.

All these are kinetic energy.

What is energy at rest:- Rest energy

E₀ = mc²

Energy is relevant to the rest mass of the particle.

The total energy of the particle is E = E₀ + K

Where K is the kinetic energy.

Also Read:

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Potential energy:- Bodies can do work by virtue of their position or state of strain.

The energy held by a body due to its position or state of strain is called the potential energy of the body. For example, the water at the top of a waterfall can rotate a turbine when falling on it. The water has this capability by virtue of its position ( at height ).

Gravitational potential energy:- Suppose a body of mass m is raised to height h from the earth’s surface. In this process, we have to do work against the downward force of gravity (mg).

Work done

W = mgh

This work is stored in the form of gravitational potential energy U. Thus,

U = W = mgh

In this expression, the gravitational potential energy on the surface of the earth is assumed to be zero. If the body falls back on the earth, an amount mgh of work can be obtained from it.

Also check-

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NCERT Physics Notes:

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