Posted by ProxyTracer 19 hours ago
There are often two ways to solve physics problems: one describing the problem with forces, the other reasoning with energy. So they look like the two faces of the same coin. Hence the question: which one is actually real?
Some quick arguments for and against
Energy:
+ converts between mechanical, chemical, thermal, radiative types, and even mass
+ quantum particles, when interacting, exchange energy
- looks like an integrative quantity (in the sense of mathematical integral)
Force:
+ feels very real, when you receive a ball in your face
+ we talk of fundamental forces, not fundamental energy
+ explains momentum, deformation well
- my physics teacher used to say "nobody ever saw a force"
- force is undistinguishable with acceleration
- at the quantum level forces are actually particles interacting
- at the quantum level, the uncertainty principle makes the newtonian force pointless (pun?): seems like we could know the vector's origin or the direction but not both
Think of simple notion. Why more energy is needed to accelerate moving object compared to still?
Kinetic energy possesed by any object is equal to work/effort needed to be made by an external force to accelerate it from present state to stated velocity.
If object is already moving, and i am that external force, first i had to catch up with that object, for that i had to do work make effort until i am moving at same speed as object, even after catching up, at the momennt if i try to push object, i am distracting myself engaging into 2 activity maintaining my speed same as object and trying to push so that will definetely reduce my speed, so i first had to gain slighly more speed than object before i give it a push and transfer all my momentum to object so it accelerates.
Thus i needed more effort or work to do, to accelerate moving object compared to stationary one.
That work done is kinetic energy object posses when it was accelerated from 1 to 2 and its more than when moving object from 0 to 1.
That simply explains the fact. Now how much more energy triple or quadraple that comes down to practical established formulas.
In my understanding OP was confused as when talking about,op was simply thinking if object is already moving it would take less force to move it as it already has gain momentum against all odd of nature and resistive forces, so now only work needed is to accelerate it and it doesn't include loss against resistive forces.
But to accelerate moving object the applier of force whether human or another object also needs to catch up
If one starts with Newton's 2nd law (F=ma) assumed, then one can derive kinetic energy to be 0.5mv^2, and this is what most of the answers are explicitly or tacitly doing.
One could however start with Lagrangian formulation along with KE = 0.5mv^2 and drive F=ma. This is where one needs an explanation for why KE = 0.5mv^2, and the first answer (@Ron Maimon) is providing an explanation.
Most books I have come across on Lagrangian formulation secretly assume Newton's laws.
In my opinion, Lagrangian formulation can proceed without Newton's and without even defining momentum as mv, however, now needs KE = 0.5mv^2.
There is some rotation invariance hidden in the velocity physics because you can rotate the velocity vector of an object without having to spend energy (The force you need to apply is perpendicular to the velocity so does no work).
The typical example is you have a ball fall 1m vertically, then have a 90° bend which convert the vertical velocity into horizontal velocity and no vertical velocity, then the ball fall again 1m vertically and have its vertical velocity increased by the same amount as for the first meter. You can then add a 45° degree bend ramp to redirect the ball so that it only has horizontal velocity, and have the ball fall again. For the third bend ramp the incoming velocity will have 2 units horizontal, and 1 unit vertical (I'll let you compute the appropriate angle). A fourth ramp would be 3 units horizontal and 1 unit vertical.
Because we can do this adding velocity in a perpendicular way trick we must then use Pythagoras.
Not sure how I reconcile that for systems with linear symmetry that don't admit a sphere such as a 1D harmonic oscillator (i.e. a spring). You're confusing the fact that spheres require quadratics but quadratics are not sufficient to admit a sphere.
Kinetic energy
E = (m * v^2)/2 + (3*m * v^4 )/8*c^4 + (5*m * v^6)/16*c^6 …
and so on, so kinetic energy increases infinitely faster than speed, thus it impossible to reach c, because it requires infinite amount of kinetic energy.
Why? Because of rules of wave propagation.
For instance we know that the life forms grow via cell division, but no text can address the question of "why". They can only talk about "how".
Infact, science quest is not really about answering "why" all the way down the causal chain. It is about learning how the qualities of things are related and a bit of shallow causal chain inspection.
The causal chain, by nature, does not allow full inspection. It's dependency on temporal constructs means it breaks down where time breaks down. Infact causality might might break down at macro levels as well, leading to loops with no end or beginning (kind of chicken and egg problem).
But as others have mentioned this is only as intuitive as F=ma, or p=mv.
In my view, at least classically it's just a matter of definitions then. If our definitions of energy differ, the only thing we will experimentally agree on is the equation of motion, and even then up to a frame transformation.
∑ⱼ mⱼ v⃗ⱼ = 0⃗
where the mⱼ are the masses of the parts of the object and the v⃗ⱼ are the velocities of those parts.
If the object initially has 0 velocity, its kinetic energy is:
T = ½∑ⱼ mⱼ v⃗ⱼ²
Now we give the object a kick (or just switch reference frames) to change its velocity by Δv⃗. The new kinetic energy is:
T' = ½∑ⱼ mⱼ (v⃗ⱼ + Δv⃗)²
T' = ½∑ⱼ mⱼ (v⃗ⱼ² + 2v⃗ⱼ⋅Δv⃗ + Δv⃗²)
T' = ½(∑ⱼ mⱼ v⃗ⱼ²) + Δv⃗⋅(∑ⱼ mⱼ v⃗ⱼ) + ½Δv⃗²(∑ⱼ mⱼ)
If M is the total mass of the object, then we can substitute this into the sum in the last term. And we already saw that the sum in the middle term was 0. So:
T' = ½(∑ⱼ mⱼ v⃗ⱼ²) + Δv⃗⋅0⃗ + ½Δv⃗² M
T' = ½∑ⱼ mⱼ v⃗ⱼ² + ½MΔv⃗²
So in terms of the original kinetic energy T, which was purely thermal energy, we get:
T' = T + ½MΔv⃗²
In other words, because of the quadratic kinetic energy formula, we can see that the total kinetic energy T' of a hot object is just its thermal kinetic energy T plus the usual mechanical kinetic energy ½MΔv⃗².
T = ½∑ⱼ mⱼ v⃗ⱼ²