What is kinetic energy?

Question

I am asking these question on a fundamental level please don't go too deep in explaining. A very basic definition if energy is that It is the capacity to do work, my question is "on what thing this work is done" is the work done on some other abject or on itself? for example if an object has x amount of kinetic energy then that means;

1.x amount on work has been done on that body there it possesses x amount of energy

2.That object can do X amount of work when it collides with another object

Same goes for potential energy for example if a body has Y amount of potential energy then that means that body can convert that potential to y amount of kinetic energy when set free

Now these are the definitions that i formed myself from reading textbooks, i want you to verify these if they hold true and also answer my first question i genuinely need your help please don't go to quantum mechanics or tell me about punctuation errors please

Answer 1

"is the work done on some other abject or on itself?"

(2) is clearer. For example I might catch a ball thrown at me, and the ball would do work against my hands and arms as I bring it to rest. I acquire extra thermal energy equal to the kinetic energy lost by the ball. [A purist might say that the work is done by the force that the ball exerts on me, as it pushes my hands through a distance as the ball comes to rest.]

(1) doesn't uniquely define kinetic energy. I could do work on a body by rubbing it with a cloth. This would increase the body's internal (thermal energy) but wouldn't give it (macroscopic) kinetic energy. But we could close this loop-hole by defining the amount of a body's kinetic energy as the work done on it by a force (from another body) acting on the body (initially at rest) and pushing it, or pulling it, through a distance. This assumes that no other forces are acting on the body.

Note that I wrote "the amount of a body's kinetic energy" rather than just "a body's kinetic energy". To say that the work done on the body (even with my amendments above) is the kinetic energy that it acquires doesn't give you much of an idea as to what kinetic energy IS. Your (2) gives you a better idea.

Answer 2

Basically energy is a quantity that is conserved through time. If the energy is the sum of two terms: $$ E = U + K $$ Then it is obvious that the total value of $E$ will be divided between $U$ and $K$. If one loses energy the other one has to gain it and viceversa. Is that simple. Note that in the case of a collision, it has to be elastic, if not, energy would be lost via heat (and your equation wont work).

Work (usually denoted with $W$) is defined as quantity that measures how much a given displacement on an object goes with a force that is being applied to it. If your displacement goes in the direction of the force then: $W > 0$, if it is in the opposite direction: $W < 0$, and if it is perpendicular: $W = 0$. Work is done by the object following a path in which it suffers a force.

Potentials arise when your force is conservative (such as gravity and electrostatic forces). By the definition given before, you could interpret the potential as how much work you should do, to move a particle from a given point to $r \to \infty$ following that conservative force (only in the case of central forces, with and inverse relation).

Here is an example to understand it better. In the case of a particle moving from $r_1 \rightarrow r_2$ under the influence of gravity:

$$ W_{1\rightarrow 2} = U(r_1) - U(r_2) $$

$$ U(r) = -\frac{GMm}{r} $$

Case 1: $r_1 < r_2$ $$ W_{1\rightarrow 2} = U(r_1) - U(r_2) $$ because $U(r) \propto -1/r$ it is obvious that: $$ U(r_1) < U(r_2) $$ then: $$ W = U(r_1) - U(r_2) < 0 $$ which means negative work. If you think it, it make sense, because to reach a higer $r$ value, you need to do work against the gravitational force.

Case 2: $r_1 > r_2$ $$ W_{1\rightarrow 2} = U(r_1) - U(r_2) $$ because $U(r) \propto -1/r$ it is obvious that: $$ U(r_1) > U(r_2) $$ then: $$ W = U(r_1) - U(r_2) > 0 $$ which, again, it make sense. Positive work indicates work done with the force. If $r_2 < r_1$, that means that the particle has fall (apparently) by the influence of the gravitational force.

Case 3: $r_2 \to \infty$ $$ W_{1\rightarrow 2} = U(r_1) - U(r_2) $$ $$ U(r_2 \to \infty) = 0 $$ $$ W_{1\rightarrow \infty} = U(r_1) $$ which as said before, is the definition of the potential.

Answer 3

Reference/Citation: E. T. Whittaker "A Treatise on the Analytic Dynamics of Particles and Rigid Bodies" 2nd Edition (1917) at pages 37 - 38.

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What is kinetic energy?

In non-relativistic classical mechanics, the definition of kinetic energy for a single particle of mass $m$ is given in a cartesian coordinate system (where the particle location is given by x, y, and z) as: $$ T_1 = \frac{1}{2}m\left(\dot x^2+\dot y^2+\dot z^2\right)\;, $$ where $\dot x = \frac{dx}{dt}$ and so on, where $t$ is time.

In the same regime, the kinetic energy for a system of $N$ particles is: $$ T=\sum_{i=1}^N\frac{1}{2}m_i\left(\dot x_i^2+\dot y_i^2+\dot z_i^2\right)\;. $$

Given these definitions, one can show that the Newtonian equations of motion for the system, in an arbitrary coordinate system $\mathbf{q}$, can be re-written as the Lagrange equations of motion: $$ \frac{d}{dt}\frac{\partial T}{\partial \dot q_i} - \frac{\partial T}{\partial q_i} = F_i\;,\tag{1} $$ where the work done by the external forces in an arbitrary displacement is: $$ \delta W = \sum_i F_i\delta q_i $$

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Same goes for potential energy...

Regarding potential energy: When the forces on the system can be described in terms of a potential energy function $U$, then the work is given by: $$ \delta W = -\sum_i \frac{\partial U}{\partial q_i}\delta q_i $$ and in that case the Lagrange equations of motion reduce to: $$ \frac{d}{dt}\frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = 0\;,\tag{2} $$ where $L = T - U$.

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A very basic definition if energy is that It is the capacity to do work, my question is "on what thing this work is done" is the work done on some other abject or on itself?

The work is done on the system of $N$ particles by the external forces. The forces that are external to the system of $N$ particles can cause a change in kinetic energy. This is the content of the Work Kinetic Energy Theorem.