Energy intro, and translational kinetic energy

2025-10-26 – Dennis Evangelista

Introduction

In middle school science class you may have discussed energy and thought about how it comes in different forms. For example, when you go to the gas station and pay 5 bucks a gallon, you are buying gasoline which is chemical energy. Similarly, when you plug your laptop into the wall you are getting electrical energy. You might consider nuclear energy, or solar energy, or wind energy. You may have heard that heat is a form of energy; or that there are notions of potential energy stored by weights or springs or pressurized fluids, or kinetic energy of something moving. In biology class, they even talk about how that cheeseburger or that banana is energy.

Think a little deeper about these. How is it that these are all energy? How is liquid gasoline in a tank similar to voltage coming from the wall, or stretched molecules in a springy crossbow, or a counterweight held up high, or fissioning uranium atoms in a nuclear reactor? It seems kind of crazy that all these very different things can be thought of as the same type of physical quantity. Let’s think about that some more.

A thought experiment and some heavy lifting

A theme in physics is to be quantitative; we want to be able to measure things we care about and use computation and mathematics to gain understanding. We can use thought experiments to help shape our thinking and be precise about big concepts, so let’s do one.

S&E Crossfit

Consider two weights: a 500 barbell versus a dinky little 2 hand weight. Also, consider two cases: lifting each weight to a height of 0.5 m or lifting them fully above the head to about 2 m. Which case does the most work / requires the most energy? Which case does the least work?

First, consider the weight, or the force (\(F=mg\)) that must be exerted to lift a barbell of mass \(m\), where \(g=\qty{9.81}{\meter\per\second\squared}\). Is it more work to lift a heavy 500 barbell or a dinky 2 hand weight, all else being equal?

Next, consider the distance we must lift the weight. Does it take more work to lift the weight 0.5 m or 2 m?

So the amount of work or energy it takes to do these tasks seems to depend on force and distance over which the force is exerted. To be precise, let’s combine those into an equation \[\begin{equation} W = \text{force}\cdot\text{distance} = \vec{F}\cdot\Delta\vec{x} \label{eq:workdefn} \end{equation}\] where \(W\) is work or energy, \(\vec{F}\) is force, and \(\Delta\vec{x}\) is the distance. This combines the idea that if we exert bigger forces (as in lifting a bigger weight) we are doing more work; and that as we go longer distances we also do more work.

The units of \(W\) are N m, or alternatively, joules (J), where \(\qty{1}{\joule}=\qty{1}{\newton\meter}\). The quantity \(W\) is a scalar. Sometimes, rather than the letter \(W\), you may see the letters \(Q\), \(U\), or \(E\) used to denote energy of other types.

Kinetic energy

Now consider the case of a mass \(m\) that is moving at some velocity \(v\). How much energy does it have by virtue of its movement? Alternatively, how much work did it take to get it moving?

Not sure how to figure this out? Let’s fall back on a thought experiment. First let’s consider the energy contained by a gnat (small \(m\)) versus an 18-inch battleship artillery shell (big \(M\)). Which has more energy? Now consider two cases: one that is moving very slowly (small \(v\)) versus one that is moving very fast (large \(v\)). Which has more energy?

So we think the (kinetic) energy depends on mass \(m\) and velocity \(v\), but how? Let the units of energy give us a clue: \[\begin{align} \unit{\joule} &= \unit{\newton} \cdot \unit{\meter} \\ &=\unit{\kilo\gram\meter\per\second\squared} \cdot \unit{\meter} \\ &= \unit{\kilo\gram\meter\squared\per\second\squared} \\ &= \unit{\kilo\gram} \left( \unit{\meter\per\second}\right)^2 \end{align}\]

The units suggest that \(m\) (units kg) and \(v\) (units m s⁻¹) combine in the following way to calculate kinetic energy: \[\begin{equation} KE = f(mv^2). \end{equation}\]

The actual formula for kinetic energy is \[\begin{equation} \text{kinetic energy}\ KE = \frac{1}{2} mv^2 \end{equation}\] where \(m\) is mass and \(v\) is velocity; \(v^2\) is the magnitude of the velocity \(\vec{v}\) squared. It is a scalar, and its units are J.

Kinetic energy represents the energy something has based on how massive and how fast it is moving. Because of conservation of energy, it also represents how much work it took to accelerate the mass to its current speed; as well as how much energy must be dumped in order to bring the mass to a stop.

Example

Xuan the Guide Dog Puppy (mass 65 , 29.4 kg) is in the park when he is given the recall command come!. He begins running back towards his handler at 8 m s⁻¹. What is Xuan’s kinetic energy?

\[\begin{align} KE &= \frac{1}{2}m v^2 \\ &= 0.5 \cdot \qty{29.4}{\kilo\gram} \cdot (\qty{8}{\meter\per\second})^2\\ &= \qty{941}{\joule} \end{align}\]

Definition

An object’s translational kinetic energy is given by the equation \[\begin{equation} KE = \dfrac{1}{2} m v^2 \end{equation}\] where \(m\) is mass in kg and \(v\) is velocity in m s⁻¹. Translational kinetic energy is a scalar quantity measured in joules (J). We will see later objects can also have rotational kinetic energy which we will ignore for now.

As we stated, kinetic energy is the energy a mass has by virtue of it moving fast. A pebble sitting motionless has very little energy, while a massive container ship heading full speed at a bridge pier might have a lot of energy, so we expect the quantity to vary based on both \(m\) and \(v\) in a way that when both are big, kinetic energy is big. You might ask, why not do \(mv\)? The answer is that we do, but \(\vec{p}=m\vec{v}\) is an entirely different quantity called momentum, which we will deal with later.

Note that different observers may measure different values of the translational kinetic energy of an object, depending on the observer’s frame of reference.

Energy in general

Energy is a hugely important concept in physics, chemistry, and biology.

In physics, we briefly mentioned gravitational potential energy before, the energy something has by virtue of how high it is or where it is in a gravitational field. In your other classes, you may have heard about other types of energy, like thermal energy, nuclear energy, electrical energy, chemical energy, etc... A mind-blowing thing about physics is that all of these are energy; they’re all measured in the same units (J) and they all reflect the potential some system has to do work. Exactly how boring a cannon, lifting some weights, burning some fuel, connecting a battery are all fundamentally the same physical thing we will discuss more as we go.

Conservation of energy

You probably have also heard of conservation of energy and the First Law of Thermodynamics, which students often know as

Energy is neither created nor destroyed, it only changes form.

A version in the form of an equation as often discussed in introductory physics and engineering thermodynamics class is \[\begin{equation} \Delta U = Q - W \end{equation}\] where \(U\) is the total energy, \(Q\) are heat transfers and \(W\) are work transfers.

We will consider conservation of energy in a bit. Conservation laws are important in physics; they tell us some quantity is expected to be constant and so we get equations we can use to quantitatively describe or understand the world.