Intuitively Understanding Work and Energy It is easy to understand the concepts of momentum and impulse. The formula $mv$ is simple, and easy to reason about. It has an obvious symmetry to it.
The same cannot be said for kinetic energy, work, and potential energy. I understand that a lightweight object moving at very high speed is going to do more damage than a heavy object moving at a slower speed (their momenta being equal) because $E_k=\frac{1}{2}mv^2$, but why is that? Most explanations I have read use circular logic to derive this equation, implementing the formula $W=Fd$. Even Samlan Khan's videos on energy and work use circular definitions to explain these two terms. I have three key questions:


*

*What is a definition of energy that doesn't use this circular logic?

*How is kinetic energy different from momentum?

*Why does energy change according to $Fd$ and not $Ft$?

 A: 
What is a definition of energy that doesn't use this circular logic?

Historically, people had no clue that energy was conserved, basically because it's not obvious that when mechanical energy appears to dissipate at least partially into nothingness, actually it's turning into heat. Often the temperature changes involved are very small and not noticeable. But people had a clear intuitive idea that $Fd$ was a good figure of merit for what was being done by a horse or a steam engine, so they called it work. Later, when conservation of energy was discovered, they had this preexisting numerical scale, and they realized that it was a measure of the transfer or transformation of energy, so they started using it as the unit of energy.
From a modern point of view, there is another, nicer way to proceed. We start with some more fundamental definition for energy. For example, we can define some standard form of energy such as kinetic energy. Then, exploiting and constrained by conservation of energy, we determine a numerical scale for this form of energy and for other forms of energy that can be converted to and from it, such as gravitational potential energy. The Feynman quote in TreyK's answer is a presentation of this philosophy. One can then define work in terms of energy, as the amount of energy transferred by a macroscopic force, and prove a theorem that it's measured by $W=Fd$ under certain conditions. Or we can stick with $W=Fd$ as a definition of work, in which case we can prove as a theorem that it equals the energy transferred.

[...] $E_k=\frac{1}{2}mv^2$, but why is that?

The factor of 1/2 in front is purely a historical artifact. Conservation laws don't change their validity when you change units, so we could have any factor out in front that we liked. But if, for example, we chose to define kinetic energy as $mv^2$, then we'd have to change the numerical factors in every other equation relating to energy, e.g., we'd have $W=2Fd$.
The proportionality to $m$ has to be that way because conservation laws are additive. E.g., if KE was defined as $m^2v^2$, it wouldn't be additive when you added the energies of two different objects.
The factor of $v^2$ doesn't have to be that way logically, and in fact it isn't really $v^2$ -- relativistically the correct equation is different, and $v^2$ is only an approximation for velocities that are small compared to the speed of light. However, if we assume Newtonian mechanics to be a good approximation, then it does have to be $v^2$. There are various ways of proving this. For example, in Newtonian mechanics momentum equals $mv$ and is conserved. If you take KE to be proportional to $v^2$ and also want energy to be conserved regardless of your frame of reference, then you get a condition that is exactly the conservation of $mv$. For any other proportionality besides $v^2$, the behavior of the conservation laws for energy and momentum would not be consistent with each other when you changed frames of reference.
A: You may want to see Why does kinetic energy increase quadratically, not linearly, with speed? as well, it's quite related.
Mainly the answer to your questions is "it just is". Sort of.

What is a definition of energy that doesn't use this circular logic?

Let's look at Newton's second law: $\vec F=\frac{d\vec p}{dt}$. Multiplying(d0t product) both sides by $d\vec s$, we get $\vec F\cdot d\vec s=\frac{d\vec p}{dt}\cdot d\vec s $
$$\therefore \vec F\cdot d\vec s=\frac{d\vec s}{dt}\cdot d\vec p$$
$$\therefore \vec F\cdot d\vec s=m\vec v\cdot d\vec v$$
$$\therefore \int \vec F\cdot d\vec s=\int m\vec v\cdot d\vec v$$
$$\therefore \int\vec F\cdot d\vec s=\frac12 mv^2 +C$$
This is where you define the left hand side as work, and the right hand side (sans the C) as kinetic energy. So the logic seems circular, but the truth of it is that the two are defined simultaneously. 

How is kinetic energy different from momentum?

It's just a different conserved quantity, that's all. Momentum is conserved as long as there are no external forces, kinetic energy is conserves as long as there is no work being done.
Generally it's better to look at these two as mathematical tools, and not attach them too much to our notion of motion to prevent such confusions.

Why does energy change according to $Fd$ and not $Ft$?

See answer to first question. "It just happens to be", is one way of looking at it.
A: After more digging, I came up with this quote from Feynman -

There is a fact, or if you wish, a law governing all natural phenomena that are known to date.  There is no known exception to this law – it is exact so far as we know.  The law is called the conservation of energy.
It states that there is a certain quantity, which we call “energy,” that does not change in the manifold changes that nature undergoes.  That is a most abstract idea, because it is a mathematical principle; it says there is a numerical quantity which does not change when something happens. 
It is not a description of a mechanism, or anything concrete; it is a strange fact that when we calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same.
It is important to realize that in physics today, we have no knowledge of what energy “is.”  We do not have a picture that energy comes in little blobs of a definite amount.  It is not that way.  It is an abstract thing in that it does not tell us the mechanism or the reason for the various formulas.

As Manishearth's answer demonstrated, it is certainly possible to show the mathematical principles that go into understanding energy, but it seems to me to be a formula meant for mathematical convenience (as is Torricelli's equation), and not something meant to be intuitively understood in and of itself -

Generally it's better to look at [kinetic energy and momentum] as mathematical tools, and not attach them too much to our notion of motion to prevent such confusions.

A: kinetic energy by its name is energy of motion of a mass as opposed to e.g. potential energy, electrical energy, heat energy, etc.
The easy geometrical explanation of $Ke= 1/2 mv^2$ is drawing the right triangle of MV, momentum as vertical side and V as the horizontal side. the area of right triangle represents the total Ke energy while it is being  gradually converted to another kind of enrgy, or if we integrate MV along the axis of V: $\int MV.dv|= 1/2 MV^2$ !
A: "Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone." (Einstein and Infeld, The Evolution of Physics)
I had a similar question as OP about energy while going through Dave Farina's course on Classical physics (https://youtube.com/playlist?list=PLybg94GvOJ9HjfcQeJcNzLUFxa4m3i7FW).
What actually are energy and work? What features of reality are we talking about when we use these words? I don't think it's enough to say that they are useful but arbitrary definitions. And I think we can do better than saying than energy is not to be intuitively understood in and of itself. A definition is useful because it picks out some relevant aspect of nature, something real that our words correspond to. That's how our words have meaning. If we refuse to refer to intuition, we lose this meaning and just use formulas by rote. Aspects of nature can be intuited in our experience, and we use scientific concepts to represent and analyze them. So what aspects of nature are represented by work and energy? After some reflection, here are my answers to OP:

*

*Energy is acceleration that has been materialized and spatialized, or simply spatialized force.

*Momentum is velocity that has only been materialized. Unlike energy, the spatialization step is not carried out, nor is there a change in the velocity.

*To spatialize force, we must multiply $F$ by a spatial length, $d$. Multiplying by $t$ extends force in time instead of space. But this only cancels out one of the two divisions by $t$ that we've already done to get from displacement to velocity to acceleration. It brings us back from $ma$ to $mv$ and so $F \Delta t = \Delta mv$.

See below for the full explanation. This is basically an extended version of dimensional analysis with some interpretation. I'll first stimulate intuition using an analogy from the way kinematics becomes dynamics. Then I'll give a definition of energy from first principles using as little math as possible.  Finally I'll answer the 3 questions more fully. I think the key is to use analytical imagination to form intuitive concepts of physical quantities and their combinations.
From kinematics to dynamics
The key move in dynamics is the introduction of mass as a quantity. Kinematics discusses displacement, velocity and acceleration, but it abstracts them from the matter of the objects involved. We include the dimension of mass by multiplying each quantity from kinematics by $m$:
$$d \rightarrow m \cdot d\\v \rightarrow m \cdot v\\a \rightarrow m \cdot a\\$$
This is what Newton did when he referred to momentum as a meaningful quantity measured in kilograms-meters per second that combines both the rate of motion and the quantity of matter of an object. In the same way, force takes account of both the mass and the acceleration of an object, not just the acceleration as in kinematics. I don't know if $m \cdot d$ has a name, but we could call it something like "material extension" or a "length of matter".
In effect the kinematic concepts are made more concrete by including the factor of mass, which is a concrete reality in nature that we know by intuition (i.e. by the seeing/feeling that matter has resistance). We can therefore call this procedure of introducing mass the materialization of displacement, velocity and acceleration. $d, v, a$ are abstract concepts in kinematics, and we make them less abstract by including mass alongside them. That's how we get from kinematics to dynamics.
'Spatializing' kinematics
Now let's take the above procedure, but instead of introducing the dimension of mass let's introduce the spatial dimension. We spatialize displacement, velocity and acceleration by including the spatial concept of displacement, distance or length. We do this by multiplying each by $d$. We get:
$$d \rightarrow d \cdot d\\
v \rightarrow d \cdot v\\
a \rightarrow d \cdot a\\$$
The first is a "length of a length", or simply area, measured in $m^2$. We can call the second a "length of motion" by analogy with Newton's "quantity of motion" for $mv$. Here we want to imagine a single spatial dimension that is not empty (as $d$), or filled with matter (as $m \cdot d$) but "filled" with motion ($d \cdot v$). Finally we have a "length of acceleration", $d \cdot a$, which is a space that "contains" acceleration and nothing more. Our imagination can make these abstract combinations, even though we never encounter a "length of motion" or a "length of acceleration" as separate realities in experience. Acceleration is always of some mass, in some specific context, etc. But in science we abstract away to focus on separate elements.
Work and energy
Based on the formula $W = F \cdot d$ and the above discussion we can give the following definition of work:
Work is spatialized force.
To multiply $F$ by $d$ just means to extend force in space, or to 'spatialize' it. More fully we can say that work is spatialized and materialized acceleration, which is evident after simple replacement: $W = m \cdot a \cdot d$. When work is done, acceleration is 'combined' with mass on the one hand, and with distance on the other. Work thus produces a length of force, or a length of a material quantity of acceleration. We could also say that work actualizes a force in space by taking account of the length of the space over which the force is applied.
We get to kinetic energy by working with the formulas. Assuming initial velocity of $0$ and constant $a$:
$$d = {1 \over 2}vt \text{ , } a = {v \over t}$$
Therefore spatialized acceleration from above reduces to:
$$d \cdot a = {1 \over 2}vt \cdot {v \over t} = {1 \over 2}v^2$$
Then we introduce mass to get acceleration that is both spatialized and materialized, or work, which is equal to the change in kinetic energy:
$$d \cdot a \cdot m = W = {1 \over 2}mv^2 = E_k$$
To answer OP's questions:

*

*Energy can be defined without circularity as spatialized and materialized acceleration, or simply as spatialized force, measured in $Nm$ or Joules. This only refers to our intuitive concepts of space, matter/mass and acceleration. (Acceleration in turn refers to the concepts of change, space and time.) It's true that we start with the formula $W=mad$, and you could say that we group m, a and d by arbitrary choice. But this grouping refers to an aspect of concrete reality, and that's what a definition expresses. It's not enough to just give symbols and logical operations. Our physical concepts actually refer to nature that is outside of us.

*Energy is acceleration that has been materialized and spatialized. Whereas momentum is velocity that has only been materialized ($mv$). The spatialization step has not been carried out, nor is velocity changing. If we spatialize momentum, we will get a length of momentum, or $mvd$. If we then take time rate of change of its velocity, we will get work or $mad$. We can say that momentum is the constant motion of a mass, whereas energy is the acceleration of a mass that has been extended in space.

*$F \cdot d$ lets us 'spatialize' force: it represents the extension of force in space. The reality of that Force-space is what we mean by energy. On the other hand $F \cdot t$ 'temporalizes' force or extends it in time. However, we've already divided by time twice to arrive from distance to velocity and from velocity to acceleration (hence the units of force are $kg \cdot {m \over s^2}$). So the t in $Ft$ will cancel one of these divisors to give us $mv$, and thus $Ft$ is impulse or the change in momentum.

A: The only things in physics about which we can be certain are velocity, distance, and acceleration. The rest are just abstract concepts such as force, momentum, energy, etc. The entirety concepts of work and energy revolves around the third equation of motion.
Work and energy are introduced in physics to solve problems only. They don't have any use per se. It is easy to analyze the motion of an object with the help of work and energy concepts rather than recalling equation of motion. To get an intuitive idea of the importance of the concept of work energy theorem you have to think like a theoretical physicist back in 19th-century. You have to understand what theoretical physics really is. Developing an understanding of such abstract concepts require knowledge of the history of physics. For that use this link
