kinetic energy

Kinetic energy is the energy of motion. The motion itself may involve the movement of a body from one position to another (translational kinetic energy), the vibration of, for example, a spring (vibrational kinetic energy) or the rotation of a body about its own axis (rotational kinetic energy).

The strict definition of kinetic energy is that it is the work required to get an object to move (in the translational case) with a velocity v. Given a force F exerted on a body of mass m over a distance ds, the work done is given by

W= ∫F·ds

Noting Newtons second law (F=m dv/dt) and the definition of velocity (v=ds/dt) the equation may be rewritten.

W=m∫ dv/dt·(vdt)

The nature of differentiation allows one to write

dv/dt.v=(1/2)d(v.v)/dt=(1/2)dv²/dt

Substituting this into the above yields a simple expression for the translational kinetic energy

(1/2)mv²

Similarly, the rotational kinetic energy of a body with moment of inertia I and angular velocity ω is given by

(1/2)Iω²

When driving at 30 m.p.h an increase of speed to 33 m.p.h leads to an over 20% increase in kinetic energy and a corresponding increase in the chance of serious injury or death if there were an accident at the increased speed (not 10% as might be supposed by someone who supposes a linear relationship between speed and energy). Note, a similar relationship applies between the stopping distance of a car and its speed (thanks trikyguy).

The temperature T of a body is proportional to the average translational kinetic energy of its atoms or molecules.

<(1/2)mv²>_average=(3/2)kT

where k is the Boltzmann constant

In the special theory of relativity, there is a new expression for kinetic energy. The difference between the relativistic expression and the classical expression does not come from any conceptual difference in the idea of energy. Kinetic energy still is the amount of work required to move an object.

To derive the expression for relativistic kinetic energy you firstly define force, which is defined as the change of momentum over time.

F ≡ dp/dt

Relativistic momentum is given by the expression p = γmv. Where γ represents the Lorentz Transformation gamma factor.

γ = (1-^v2⁄_c²)^-1/₂, c represents the speed of light in a vacuum.

Substituting relativistic momentum into the equation which defines force gives:

F = d(γmv)/dt

Kinetic energy is correctly defined in the write-up by Blush Response above, it is:

E_k = ∫Fds = ∫d(γmv)•ds/dt = ∫γmvdv

Leaving out some of the complicated algebra, the solution to the relativistic exrpession for kinetic energy is:

E_k = mc²(γ - 1)

Expanding γ with the binomial theorem gives: γ = 1 + 0.5• ^v2⁄_c² + ...

When substituting into the relativistic equation of kinetic energy, you approach the classical result 0.5•mv² when speeds are much smaller than the speed of light.

At large velocities the differences are significant. Classically if you are travelling at 80% the speed of light, and increase to 95% the speed of light, it requires an increase of 41% more energy. Relativistically, to increase from 80% of the speed of light to 95% of the speed of light, it requires 92% more energy, over twice as much as predicted by the classical theory.

With all due respect to the above authors, their writeups are scary. They use funny symbols like ∫ and weird phrases like "theory of relativity." Please, where are the roller coasters? How can you have a discussion about kinetic energy without roller coasters? Everybody can understand roller coasters.

Kinetic energy is the energy of a moving object, and is dependent on the mass of the object and the velocity at which it is traveling. Heavier objects and faster moving objects have more kinetic energy — that is to say, a semi-truck traveling at highway speeds has more kinetic energy than a baby carriage traveling at walking speed. You could stop a baby carriage with your hand, but a semi-truck would barely slow down at all as it ran you over because it has a lot more energy. Kinetic energy is the energy required to bring an object up to a velocity, or to stop the object which is traveling at that velocity.

As explained above in the other writeups, the equation for calculating the kinetic energy of a moving object is equal to one half of an object's mass times the square of its velocity.

E_K = ½mv²

This is the only kinetic energy equation needed by everyone except the people who are about to tell me it's not the only one they need, and those people already know the other ones.

Because the velocity term is squared, changing the velocity makes a bigger difference than changing the mass. Doubling the mass doubles the kinetic energy, but doubling the velocity quadruples the kinetic energy.

Kinetic energy is only one kind of energy. Energy comes in several different forms, such as chemical energy, gravitational potential energy, the stress stored in a compressed spring, and many others. In many cases, potential energy is stored for later conversion into kinetic energy. A bullet, for example, stores chemical potential energy in its gunpowder. When the bullet is fired, a lot of chemical energy is released in a very short amount of time, which causes a rapid expansion of gas and propels the bullet out the barrel. In this case, m is the mass of the bullet and v is the velocity of the bullet. Using either a heavier bullet or more gunpowder could increase the kinetic energy.

Chemical energy conversion is usually one-way only; once the energy has been released from the chemical, a very difficult and complex process is usually required to put it back into chemical form.

Gravitational energy on the other hand is easily converted back and forth between potential and kinetic. A pendulum, for example, converts energy back and forth from potential to kinetic every half of a swing. When the pendulum is at its highest point, it stops moving for an instant as it changes direction. When it is at this point, all of its energy is gravitational potential. As the pendulum begins to swing down, some of its gravitational potential energy is converted into kinetic energy. The farther down it swings, the more gravitational energy is converted to kinetic energy, and the faster the pendulum goes. When the pendulum is at the very bottom, it is moving at its maximum speed, and all of its available gravitational energy has been converted to kinetic. It then begins to swing back up, converting kinetic energy into gravitational potential energy and slowing down until it reaches its highest point, and the cycle begins again.

But a pendulum slows down, and eventually comes to a stop. This is due to efficiency losses in the system: the pendulum weight must push against air resistance and there is friction in the pendulum's pivot. Efficiency losses waste energy, which is then no longer available to the system, so at the end of every swing there is slightly less gravitational potential energy available — that is, the pendulum swings up slightly lower each time.

Say for example a 1 kilogram pendulum weight is lifted up 1 meter. It now has gravitational potential energy equal to its mass times the pull of gravity times the height to which it is lifted:

E_P = mgh
(1 kilogram)(9.81 meters per second squared)(1 meter) = 9.81 kilogram meters squared per second squared, or Joules

When the pendulum is released, the weight swings down under the force of gravity: energy is converted from potential energy to kinetic energy. At the bottom of its swing, it has reached its full speed as all of its available potential energy has been converted to kinetic energy — that is, the 9.81 Joules of potential energy are now 9.81 Joules of kinetic energy.

E_K = ½mv² → v = √(2E_K/m)
The square root of (2 × 9.81 Joules ÷ 1 kilogram) = 4.43 meters per second

However, this is only the case without considering efficiency losses! If we could measure the speed of the pendulum at the bottom of the swing, we could use the actual speed to calculate how much kinetic energy there actually was, and the difference between this result and the potential energy calculated above would be the amount of energy lost to friction and air resistance.

Let's say we measured the speed at the bottom of the swing to be 4.33 m/s. The efficiency losses are then E_P - E_K, or

efficiency loss = E_P - ½mv²
9.81 Joules - (½)(1 kg)(4.33m/s)² = 0.44 Joules

We could also do this by measuring how high the pendulum swings back up on the other side to calculate how much potential energy was left after a full swing was completed, and subtracting it from how much potential energy was available at the beginning. But keep in mind that the kinetic energy calculation was the energy lost in half a swing, and the potential energy calculation would be the energy lost in a full swing.

But I promised you roller coasters:

Roller coasters are typically designed with a very big hill at the beginning and several smaller hills along the track. A chain or hoist of some sort is used to pull the roller coaster up to the top of the first hill. The hoist uses energy to do this, energy which is being converted into gravitational potential energy as the roller coaster is lifted higher and higher. At the peak of this first hill, the roller coaster has all of the energy that it will have for the entire ride, stored as gravitational potential energy.

This is a very important point, since the roller coaster cannot pick up more energy during the ride (in most designs) and is in fact losing energy during the ride due to friction and air resistance losses. This means that every other hill on the ride must be smaller than the first hill was, or the roller coaster won't be able to climb them.

The first thing the roller coaster does on its own is plunge down the first big hill. This hill usually goes from the highest point on the ride to the lowest point on the ride to give the riders the maximum speed right away. At the bottom, all of the potential energy is converted to kinetic energy, and the ride is going the fastest it can possibly go. Enjoy it, because due to friction losses the roller coaster will never again be able to quite hit that same speed.

If that was all that happened, this would be a pretty dull roller coater, so there will be a hill coming up next. This hill must be shorter than the first hill, as explained earlier. The taller this hill is, the more energy will be converted to potential energy and the more the roller coaster will slow down, so the roller coaster goes faster over small hills than it does over large hills.

As the ride continues, the roller coaster goes over several more hills, large and small, higher and lower, slowing down and speeding up — but always maintaining approximately the same amount of total energy. A small amount of energy is lost due to friction and air resistance, but it isn't much. When the ride comes to the end, it will still be going very fast since it is back at a height of zero meters — all of its available energy is currently kinetic energy. If we measured the speed at this point, and compared the kinetic energy to the potential energy granted by the first hill, we could calculate the total efficiency losses of the ride.

The ride finally comes to a stop when the operator pulls a lever, engaging the brake system for the roller coaster. Brakes provide extra friction to a system in order to waste energy intentionally. When all of the energy has been lost, the ride has come to a full and complete stop, the operator releases the harnesses for the passengers, and they disembark to free up the cars for the next riders. The hoist will then pull the cars up the first hill again, and a new group of people will feel the exhilaration of physics in action.

Also see momentum.