Kinetic energy

What is kinetic energy?

In plain words: kinetic energy is the energy a body has because it is moving. For a nonrelativistic particle of mass m and speed v, or for the translational motion of a body whose centre of mass has speed v, the kinetic energy is

The equation NCERT refers to is its Eq. (5.2b), found by multiplying the equation of motion by m/2:

NCERT calls it a special case of the work-energy theorem: the work done on a body by the net force equals the change in this quantity. So kinetic energy is also a measure of the work a moving body can do as it is brought to rest, which is why a moving hammer can drive a nail and a flowing stream can turn a mill wheel.

Properties of kinetic energy

Kinetic energy is a scalar quantity.

NCERT Physics Part I (Class 11), Section 5.4, page 74

• It is a scalar. NCERT writes it as $\tfrac{1}{2}m\,\vec v \cdot \vec v$⁠; the dot product of the velocity with itself is v², a number with no direction. A ball moving east or west at 10 m s^⁻¹ has the same kinetic energy.
• It is never negative. The mass is positive and v² is never negative, so K ≥ 0, and this translational K is zero when the centre of mass is at rest. An extended body can still have rotational or internal kinetic energy.
• Its SI unit is the joule. One joule is the kinetic energy of a 2 kg body moving at 1 m s^⁻¹. Its dimensions are [M L^² T^⁻²], the same as work.
• It grows with the square of the speed. Doubling the speed makes the kinetic energy four times as large; tripling it makes it nine times as large.
• It depends on the frame of reference. A passenger sitting in a moving train has no kinetic energy relative to the train, but a lot relative to the ground.

Kinetic energy and momentum

Here p denotes the magnitude of linear momentum, p = mv, and K is translational kinetic energy. Since v = p/m, substituting gives

so

and, the other way round, $p = \sqrt{2mK}$. Two results follow:

• For the same kinetic energy, the heavier body has the larger momentum, since p grows as $\sqrt{m}$.
• For the same momentum, the lighter body has the larger kinetic energy, since K goes as 1/m. This is why a bullet carries much more kinetic energy than the gun that recoils with a momentum of the same magnitude.

For a fixed mass, percentage changes follow from the square: if the speed rises by 10%, the kinetic energy is multiplied by 1.1² = 1.21, a rise of 21%. If the momentum doubles, the kinetic energy becomes four times as large. For small changes only, the percentage change in K is about twice the percentage change in v (or in p): a 1% rise in speed gives about a 2% rise in K. For large changes use the exact square: 10% gives 21%, not 20%.

Kinetic energy around us

• Water and wind. NCERT notes that the kinetic energy of a fast-flowing stream has long been used to grind corn, and that sailing ships use the kinetic energy of the wind. Wind turbines and water turbines do the same today.
• A hammer and a nail. The moving hammer head gives up much of its kinetic energy as work done in pushing the nail into the wood; the rest becomes heat and sound.
• Road safety. A car at 60 km/h has four times the kinetic energy it has at 30 km/h, so a crash at the higher speed has far more energy to be absorbed.
• Cricket and other sports. A fast bowler’s delivery carries much more kinetic energy than a spinner’s, though both balls have the same mass.

Solved examples

Common mistakes to avoid

Quick revision summary

• Kinetic energy is the energy of motion, half the mass times the square of the speed. It is a scalar, never negative, and measured in joules.
• It is a measure of the work a moving body can do in coming to rest.
• It goes as the square of the speed, so doubling the speed gives four times the kinetic energy.
• In terms of momentum it is p^²/2m: for equal kinetic energies the heavier body has more momentum; for equal momenta the lighter body has more kinetic energy.
• It depends on the frame of reference in which the speed is measured.

Questions students ask

Why is kinetic energy ½mv^² and not mv^²?

The ½ in K = ½mv^² comes from the work needed to bring a body up to speed. A constant net force F = ma that takes a body from rest to speed v acts over a distance s = v^²/(2a), so it does work Fs = ma × v^²/(2a) = ½mv^². Put another way, while the body speeds up uniformly its average speed is only v/2, so it covers half the distance it would at full speed v in the same time.

Can a body have kinetic energy without momentum?

A single particle cannot: K = p^²/2m is zero when its momentum is zero. An extended body or a system of particles can have kinetic energy with zero total linear momentum, because individual momenta can cancel while kinetic energies add. A rigid body spinning about its stationary centre of mass is one such system. A shell that bursts at rest into fragments, or a ceiling fan spinning on a fixed axle, has zero total momentum but plenty of kinetic energy.

What is the difference between kinetic energy and potential energy?

Kinetic energy is energy of motion, K = ½mv^², and it is never negative. Potential energy is energy stored because of position or configuration, such as mgh for height or ½kx^² for a spring; its value depends on the chosen zero, so it can be negative, and it can be defined only for conservative forces. A falling ball turns potential energy into kinetic energy, and when only conservative forces do work their sum stays constant.

What is the SI unit and dimensional formula of kinetic energy?

The SI unit of kinetic energy is the joule (J), equal to 1 kg m^² s^⁻². Its dimensional formula, from ½mv^², is [M L^² T^⁻²], the same as work and every other form of energy.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Find the kinetic energy of a 0.5 kg ball moving at 4 m s^⁻¹.

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   $$K = \tfrac{1}{2}(0.5)(4)^{2} = 4\ \text{J}$$

2. Q2. A body of mass 2 kg has kinetic energy 25 J. Find its speed and its momentum.

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   Setting $\tfrac{1}{2}(2)v^{2}$ equal to 25 J gives v² = 25 and v = 5 m s⁻¹. The momentum is 2 × 5, that is 10 kg m s^⁻¹.

3. Q3. The kinetic energy of a body increases by 300%. By what percentage does its momentum increase?

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   An increase of 300% makes the kinetic energy 4 times as large. Since $p = \sqrt{2mK}$, the momentum becomes $\sqrt{4} = 2$ times as large: an increase of 100%.