Kinetic energy is the energy that an object possesses because of its motion. It depends on the mass m and the speed v according to Ek = ½ m v², is expressed in joules (J), and it is a scalar quantity, always positive.
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How to measure it in class
Kinetic energy is not measured directly: one measures a speed, and deduces Ek from it. FizziQ offers two approaches.
By video analysis (the most visual):
- Film the motion of the object with a graduated ruler in the frame, for scale
- Mark the position frame by frame in FizziQ’s video analysis module
- Deduce the speed between two neighboring points, v ≈ Δx / Δt
- Calculate Ek = ½ m v² after weighing the object
- Plot Ek as a function of time and identify where the energy increases, decreases, or is conserved
By free fall and the accelerometer:
- Drop the smartphone from a known height h onto a thick cushion
- Use the accelerometer to identify the moment of release and the moment of impact
- Calculate the arrival speed v = √(2gh), then Ek
- Compare with the potential energy lost, Ep = mgh: the two should coincide, apart from friction
Scientific activities on this topic
Several FizziQ activities involve kinetic energy, on its own or within an energy balance:
- Elastic collision: billiards - verify that total kinetic energy is conserved in an elastic collision
- Pendulum: the relation a = 2gh/r - convert potential energy of height into kinetic energy at the lowest point
- Moment of inertia of a cylinder - discover rotational kinetic energy, which adds to translational kinetic energy
- Impact vibrations and seismics - observe where the kinetic energy of a crashing object goes
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The work-energy theorem
This is the central result of the first-year high school physics curriculum. In an inertial reference frame, the change in kinetic energy of a system between two instants is equal to the sum of the works of the forces acting on it:
ΔEk = Ek(final) − Ek(initial) = Σ W(F⃗)
This theorem is very convenient: it directly relates a change in speed to the forces, without going through the details of the trajectory or the time. For a free fall from a height h, the weight does work W = mgh, hence ½ m v² = mgh and v = √(2gh) - a result independent of the mass.
An energy that depends on the reference frame
This is a subtle point, and often misunderstood. Kinetic energy is not an intrinsic property of a body: it depends on the chosen reference frame. A passenger seated in a high-speed train has zero kinetic energy in the reference frame of the train, and a considerable amount in the reference frame of the ground. There is no contradiction: both descriptions are equally valid, and the work-energy theorem remains true in each, provided the reference frame is inertial.
Rotational kinetic energy
A rotating solid possesses kinetic energy even if its center of mass is stationary. It is written Ek = ½ J ω², where J is the moment of inertia and ω the angular velocity. For an object rolling without slipping, such as a cylinder on an inclined plane, the total energy is the sum of the two terms - which explains why a hollow cylinder rolls down more slowly than a solid cylinder of the same mass: a larger share of its energy goes into rotation.
A bit of history
The ancestor of kinetic energy is the “vis viva” or “living force” (mv²) introduced by Leibniz at the end of the 17th century, in opposition to Descartes’ “quantity of motion”. Émilie du Châtelet defended and spread this idea in France, drawing on ‘s Gravesande’s experiments with balls dropped into soft clay: the depth of penetration turned out to be proportional to v², not to v.
It was Gaspard-Gustave Coriolis who, in 1829, introduced the factor ½ and explicitly related the quantity ½ mv² to the work of forces, giving it its modern form. The name “kinetic energy” appeared a little later, around 1850, in the writings of William Thomson (Lord Kelvin) and William Rankine, at the time when the general principle of conservation of energy was being established.
At very high speeds
The formula ½ mv² is an approximation, excellent as long as v remains small compared with the speed of light c. In relativity, kinetic energy is written Ek = (γ − 1)mc², with γ = 1/√(1 − v²/c²). The difference becomes noticeable beyond about 10% of c, that is 30,000 km/s: this is the regime of particle accelerators, not of laboratory objects.
Seven Experiments on Gravity with your Smartphone (FizziQ blog) offers several energy balances that can be carried out in class | Unlocking the Physics of 12 Sports with a Smartphone describes situations where kinetic energy can be measured in the field.
Formula
Translational kinetic energy:
Ek = ½ × m × v²
Rotational kinetic energy:
Ek = ½ × J × ω²
Work-energy theorem (inertial reference frame):
ΔEk = Σ W(F⃗)
Speed acquired in free fall from a height h, without friction:
v = √(2gh)
where:
- Ek: kinetic energy (J)
- m: mass (kg)
- v: speed in the reference frame of study (m·s⁻¹)
- J: moment of inertia (kg·m²)
- ω: angular velocity (rad·s⁻¹)
- W: work of a force (J)
- g: gravitational acceleration, about 9.81 m·s⁻² in France
Application examples
- A 1,200 kg car at 50 km/h (13.9 m·s⁻¹) has about 116 kJ of kinetic energy; at 100 km/h, it has 463 kJ, four times as much. That is all the energy the brakes must dissipate as heat, and the reason braking distance quadruples.
- A 58 g tennis ball served at 200 km/h (55.6 m·s⁻¹) carries about 90 J, comparable to the energy of a 1 kg mass falling from 9 m.
- Wind turbines convert the kinetic energy of the wind: the available power varies as v³, because the flow of air through the blades also increases with speed.
- A flywheel stores energy in the form of rotational kinetic energy; some urban buses use one to recover braking energy.
- Airbags and the crumple zones of a vehicle lengthen the duration of the impact, which reduces the force experienced for the same kinetic energy to be dissipated.
FAQ
Q: Why is the speed squared and not simply proportional? A: Because energy is defined from the work of forces. For a constant force, W = F × d, and the distance needed to reach a speed v itself grows as v². Historically, ‘s Gravesande’s experiment showed this directly: a ball dropped from four times the height arrives twice as fast, but sinks four times as deep into the clay.
Q: Can kinetic energy be negative? A: No. Since it is written ½ mv² with m > 0 and v² ≥ 0, it is always positive or zero. Only its change ΔEk can be negative, when the object slows down.
Q: Can two observers find different kinetic energies for the same object? A: Yes, and both are right. Kinetic energy depends on the reference frame in which the speed is measured. It is not an absolute property of the object.
Q: Where does the kinetic energy of a braking car go? A: It is dissipated as heat in the brake discs and pads, as well as in the tires and the air. The total energy is conserved, but it passes from an ordered form (the overall motion) to a disordered form (thermal agitation), which cannot be fully recovered.
Q: What is the difference between kinetic energy and momentum? A: They are two distinct quantities. Momentum p = mv is a vector, conserved in all collisions. Kinetic energy is a scalar in v², conserved only in elastic collisions. This is precisely what makes it possible to distinguish an elastic collision from an inelastic one.
Related concepts
Mechanical Energy - Potential Energy - Conservation of Energy - Momentum - Elastic Collision - Inelastic Collision - Inertial Reference Frame