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

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

Edited By Vishal kumar | Updated on Jul 02, 2025 04:58 PM IST

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.

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

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 [ML²T].

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Background wave

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

kinetic energy

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.

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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-

NCERT Physics Notes:

Frequently Asked Questions (FAQs)

1. Write the expression for kinetic energy

We can derive the expression 


              K = 1/2mv²

2. Define kinetic energy and derive its formula

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

      Formula of kinetic energy


                                             K = 1/2mv²

3. What is the SI unit of kinetic energy?

The SI unit of kinetic energy is the joule.

4. What is the dimensional formula of kinetic energy?

The dimensional formula of the kinetic energy is  [ML²T-²].

5. Write the formula of gravitational potential energy

 The formula of gravitational potential energy is 


                                             U = mgh   

6. What is kinetic energy?
Kinetic energy is the energy possessed by an object due to its motion. It depends on both the mass of the object and its velocity. The faster an object moves or the more massive it is, the more kinetic energy it has.
7. What is the equation for kinetic energy?
The equation for kinetic energy is KE = 1/2 mv², where KE is kinetic energy, m is the mass of the object, and v is its velocity.
8. Why is there a "1/2" in the kinetic energy equation?
The "1/2" in the kinetic energy equation comes from the mathematical derivation using calculus. It arises when integrating the force with respect to displacement to find the work done, which equals the change in kinetic energy.
9. Does kinetic energy depend more on mass or velocity?
Kinetic energy depends more on velocity because it's proportional to velocity squared (v²), while it's only directly proportional to mass. Doubling the mass doubles the kinetic energy, but doubling the velocity quadruples the kinetic energy.
10. How is kinetic energy different from potential energy?
Kinetic energy is associated with motion, while potential energy is associated with position or configuration. An object in motion has kinetic energy, whereas an object at rest in a gravitational field or a compressed spring has potential energy.
11. Why do we use joules to measure kinetic energy?
Joules are used to measure kinetic energy because it's a unit of energy in the International System of Units (SI). One joule is equal to the energy transferred when a force of one newton acts on an object over a distance of one meter.
12. How is kinetic energy conserved in a closed system?
In a closed system with no external forces, the total kinetic energy remains constant. Energy may transfer between objects or convert to other forms like potential energy, but the total energy of the system remains the same.
13. Can an object have both kinetic and potential energy simultaneously?
Yes, an object can have both kinetic and potential energy at the same time. For example, a roller coaster car moving along the track has kinetic energy due to its motion and potential energy due to its height above the ground.
14. How does air resistance affect the kinetic energy of a falling object?
Air resistance reduces the kinetic energy of a falling object by converting some of it into heat. Without air resistance, all the potential energy would be converted to kinetic energy, but with air resistance, some energy is lost to the surroundings.
15. How does kinetic energy relate to the concept of work?
The work-energy theorem states that the net work done on an object equals its change in kinetic energy. Mathematically, this is expressed as W = ΔKE, where W is work and ΔKE is the change in kinetic energy.
16. How is kinetic energy transferred between objects in a collision?
In a collision, kinetic energy can be transferred from one object to another. In an elastic collision, the total kinetic energy is conserved. In an inelastic collision, some kinetic energy is converted to other forms, like heat or deformation of the objects.
17. Can kinetic energy be converted to potential energy?
Yes, kinetic energy can be converted to potential energy and vice versa. This conversion happens constantly in many systems. For example, as a pendulum swings, its energy continuously changes between kinetic (at the bottom of the swing) and potential (at the top of the swing).
18. How does the concept of kinetic energy apply to gases?
In gases, kinetic energy is related to the random motion of molecules. The average kinetic energy of gas molecules is directly proportional to the absolute temperature of the gas. This forms the basis of the kinetic theory of gases.
19. How does kinetic energy relate to the stopping distance of a vehicle?
The stopping distance of a vehicle increases with its kinetic energy. Since kinetic energy is proportional to velocity squared, doubling the speed quadruples the kinetic energy and thus significantly increases the stopping distance.
20. How does kinetic energy play a role in renewable energy sources?
Many renewable energy sources harness kinetic energy. Wind turbines convert the kinetic energy of moving air into electrical energy. Hydroelectric power plants use the kinetic energy of falling water. Even solar panels indirectly use the kinetic energy of photons.
21. Why is kinetic energy important in sports?
Kinetic energy is crucial in many sports. It determines the impact force of a ball, the speed of a runner, or the power of a punch. Understanding kinetic energy helps in designing equipment, improving techniques, and analyzing performance in various sports.
22. How does kinetic energy relate to the concept of power?
Power is the rate at which work is done or energy is transferred. In the context of kinetic energy, power can be thought of as how quickly kinetic energy is changing. For example, a more powerful engine can increase a car's kinetic energy more rapidly.
23. What is the kinetic energy of a photon?
While photons don't have mass in the traditional sense, they do have energy that can be considered a form of kinetic energy. The energy of a photon is given by E = hf, where h is Planck's constant and f is the frequency of the light.
24. How does kinetic energy relate to the concept of efficiency?
Efficiency in physics often involves the conversion of one form of energy to another. When discussing kinetic energy, efficiency might refer to how much of the input energy is converted to useful kinetic energy versus being lost to heat or other forms of energy.
25. Why is kinetic energy important in the design of roller coasters?
Kinetic energy is crucial in roller coaster design because it determines the speed and excitement of the ride. Designers use the conversion between potential and kinetic energy to create thrilling drops and loops, ensuring the coaster has enough energy to complete the course.
26. How does kinetic energy relate to the concept of impulse?
Impulse is the change in momentum, while kinetic energy is related to velocity squared. Both concepts are important in collisions. The impulse-momentum theorem relates to the force of impact, while changes in kinetic energy relate to the energy transferred or dissipated during the collision.
27. How does kinetic energy relate to the concept of work in inclined planes?
When an object moves up an inclined plane, work is done against gravity, increasing the object's potential energy and decreasing its kinetic energy. When moving down, potential energy is converted to kinetic energy. The total work done equals the change in kinetic energy, regardless of the path taken.
28. Why is kinetic energy important in understanding explosions?
In an explosion, chemical or nuclear energy is rapidly converted into kinetic energy of the explosion products and surrounding materials. The destructive power of an explosion is largely due to this sudden increase in kinetic energy of fragments and shock waves.
29. How does the kinetic energy of ocean waves relate to coastal erosion?
Ocean waves carry kinetic energy, which is transferred to the shore upon impact. This energy can erode coastlines by dislodging and moving sand, rocks, and other materials. The more kinetic energy the waves have, the more potential they have for erosion.
30. What is the relationship between kinetic energy and the speed of sound?
The speed of sound in a medium is related to the average kinetic energy of the particles in that medium. In gases, for example, the speed of sound increases with temperature because higher temperatures mean higher average kinetic energy of the gas molecules.
31. How does kinetic energy relate to the concept of escape velocity?
Escape velocity is the minimum speed an object needs to escape a planet's gravitational field. It's determined by equating the object's kinetic energy to the gravitational potential energy at the surface. If the kinetic energy is greater, the object can escape the gravitational pull.
32. Why is understanding kinetic energy important in the design of protective gear?
Protective gear, such as helmets or car crumple zones, is designed to absorb and dissipate kinetic energy during impacts. By understanding kinetic energy, designers can create gear that effectively converts harmful kinetic energy into less dangerous forms, reducing the risk of injury.
33. How does kinetic energy relate to the concept of terminal velocity?
Terminal velocity is reached when the upward force of air resistance equals the downward force of gravity on a falling object. At this point, the object's kinetic energy remains constant because it's no longer accelerating. Understanding kinetic energy helps explain why there's a limit to falling speed in air.
34. What is the relationship between kinetic energy and the greenhouse effect?
The greenhouse effect involves the absorption and emission of infrared radiation by gases in the atmosphere. This radiation carries energy, which can be considered a form of kinetic energy at the molecular level. Greenhouse gases trap this energy, leading to increased atmospheric and surface temperatures.
35. How does kinetic energy play a role in the function of airbags?
Airbags work by increasing the time over which a change in kinetic energy occurs during a collision. By extending the time of impact, airbags reduce the force experienced by occupants, as force is the rate of change of momentum. This helps convert the car's kinetic energy more safely.
36. How does kinetic energy relate to the concept of specific heat capacity?
Specific heat capacity is the amount of energy required to raise the temperature of a unit mass of a substance by one degree. At the molecular level, this involves increasing the average kinetic energy of the particles. Substances with higher specific heat capacities require more energy to increase their kinetic energy.
37. What is the relationship between kinetic energy and the formation of tornadoes?
Tornadoes form when there's a significant difference in kinetic energy between air masses. The rotation of a tornado represents an intense concentration of kinetic energy. Understanding the distribution and transfer of kinetic energy in the atmosphere is crucial for predicting and understanding tornado formation.
38. How does kinetic energy relate to the concept of activation energy in chemical reactions?
Activation energy is the minimum kinetic energy that reacting particles must have for a chemical reaction to occur. When particles collide, they must have enough kinetic energy to overcome the activation energy barrier. This explains why higher temperatures (which mean higher average kinetic energy) often speed up reactions.
39. Why is understanding kinetic energy important in the field of nanotechnology?
In nanotechnology, understanding kinetic energy is crucial for manipulating individual atoms and molecules. The kinetic energy of particles at the nanoscale affects their behavior, interactions, and the stability of nanostructures. It's also important in designing nanomachines and in processes like self-assembly of nanoparticles.
40. Can kinetic energy be negative?
No, kinetic energy cannot be negative. Since it's calculated using the square of velocity (v²) and mass is always positive, the result is always positive or zero (when the object is at rest).
41. What happens to kinetic energy when an object stops moving?
When an object stops moving, its kinetic energy becomes zero. The energy is typically converted into other forms, such as heat due to friction or potential energy if the object's position changes.
42. How is kinetic energy related to work?
The work done on an object equals its change in kinetic energy. This is known as the Work-Energy Theorem. If positive work is done on an object, its kinetic energy increases, and vice versa.
43. How does mass affect kinetic energy?
Mass is directly proportional to kinetic energy. If you double the mass of an object while keeping its velocity constant, the kinetic energy will double. This is why heavier objects have more kinetic energy at the same speed compared to lighter objects.
44. How does velocity affect kinetic energy?
Velocity has a more significant effect on kinetic energy than mass because kinetic energy is proportional to velocity squared (v²). Doubling the velocity of an object increases its kinetic energy by a factor of four.
45. What's the difference between speed and velocity in the context of kinetic energy?
In the context of kinetic energy, speed and velocity can be used interchangeably because kinetic energy depends only on the magnitude of velocity, not its direction. The equation uses v², which is always positive regardless of direction.
46. What is the relationship between kinetic energy and momentum?
While both kinetic energy and momentum depend on mass and velocity, they are different quantities. Momentum is mass times velocity (p = mv), while kinetic energy is half mass times velocity squared (KE = 1/2 mv²). Momentum is a vector, while kinetic energy is a scalar.
47. Why doesn't kinetic energy depend on the direction of motion?
Kinetic energy is a scalar quantity, meaning it only has magnitude and no direction. It depends on the speed of the object, not its velocity vector. This is why the equation uses v², which is always positive regardless of direction.
48. What's the difference between translational and rotational kinetic energy?
Translational kinetic energy is associated with linear motion of an object's center of mass. Rotational kinetic energy is associated with the object's rotation about an axis. The total kinetic energy of a rolling object is the sum of its translational and rotational kinetic energies.
49. How does kinetic energy relate to temperature at the molecular level?
At the molecular level, temperature is a measure of the average kinetic energy of particles. Higher temperatures correspond to greater average kinetic energy of the molecules or atoms in a substance.
50. What is the kinetic energy of an object at rest?
An object at rest has zero kinetic energy. Since its velocity is zero, and kinetic energy is proportional to velocity squared, the kinetic energy is also zero.
51. How does relativistic kinetic energy differ from classical kinetic energy?
Classical kinetic energy (1/2 mv²) is an approximation that works well for everyday speeds. At very high speeds (close to the speed of light), relativistic effects become significant, and the relativistic kinetic energy equation must be used, which gives higher values than the classical equation.
52. Why is kinetic energy important in understanding car crashes?
Kinetic energy is crucial in understanding car crashes because it determines the amount of energy that needs to be dissipated during the collision. The higher the kinetic energy, the more severe the crash. This is why speed limits are important for road safety.
53. What is the relationship between kinetic energy and force?
Force and kinetic energy are related through work. Force applied over a distance does work, which changes the kinetic energy of an object. The net force on an object determines its acceleration, which in turn affects its velocity and thus its kinetic energy.
54. What is the difference between average kinetic energy and instantaneous kinetic energy?
Average kinetic energy is the mean kinetic energy over a period of time or for a group of particles. Instantaneous kinetic energy is the kinetic energy at a specific moment. For a single object moving at constant speed, these would be the same, but for varying speeds or multiple particles, they can differ.
55. Why is kinetic energy important in understanding the behavior of gases?
In the kinetic theory of gases, the temperature of a gas is directly related to the average kinetic energy of its molecules. This explains many gas properties, such as pressure (caused by molecules colliding with container walls) and diffusion (due to the random motion of molecules).

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