# Minimum energy to accelerate a mass?

• jerromyjon
In summary, the energy required to accelerate a given mass is exponential, and it becomes measurable when the Newtonian kinetic energy is equal to the mass energy.
jerromyjon
I have a similar logical "gap" in my understanding that I still haven't resolved... which seems to be right on point with this thread if I call it "minimum amount of energy required to accelerate a given mass".

From what I know about SR, accelerating mass takes energy which increases exponentially as this mass approaches c. I just don't know at what speed this effect becomes measurable...

jerromyjon said:
I have a similar logical "gap" in my understanding that I still haven't resolved... which seems to be right on point with this thread if I call it "minimum amount of energy required to accelerate a given mass".

From what I know about SR, accelerating mass takes energy which increases exponentially as this mass approaches c. I just don't know at what speed this effect becomes measurable...
It isn't an exponential growth. The energy of a massive particle moving at velocity v is $E=\gamma mc^2$, where $\gamma=(1-v^2/c^2)^{-1/2}$. If you Taylor expand the expression for $\gamma$ you get
$$E=mc^2 +\frac{1}{2}mv^2 +\frac{3}{4}m \frac{v^4}{c^2} +...$$The first term is mass energy. The second is Newtonian kinetic energy. The third and later terms are where relativity disagrees with Newton on energy. So the effect is measurable when that term is measurable.

I don't know if there's an answer to your question in practical terms since I'm not current on the precision of energy measurement devices, and that probably depends on your application anyway. But that's the theoretical basis for an answer.

Thank you, I appreciate your response! I'm still trying to learn what is still "greek" to me, but that funky y that you all call gamma... isn't that only zero at zero velocity?

jerromyjon said:
gamma... isn't that only zero at zero velocity?

What do you get when you plug v = 0 into the formula for gamma? $$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

Ummm, If I set c=1 then its 1 but I'm not even sure about that c=1 scenario.

oh duh nevermind gamma has to be 1 to be at rest energy..

Ok. So what I meant was gamma is 1 at rest energy, which would only be at rest velocity. Suppose we do the M&M set-up with "identical" rockets in opposite directions, along with an identical complete system at a known relative velocity parallel to the launch vectors. This is where I'm unsure how to predict how this system evolves, but now I recall something about the relativistic "tether" problem, where the distance changes, severely complicating this visually. I'm going to play with that interactive Minkowski diagram (Thanks Ibix!) and see if I can figure it out...

## 1. What is the definition of minimum energy to accelerate a mass?

The minimum energy to accelerate a mass is the amount of energy required to increase the velocity of an object from rest to a certain speed. It is the minimum amount of energy needed to overcome the inertia of the object and initiate its motion.

## 2. How is the minimum energy to accelerate a mass calculated?

The minimum energy to accelerate a mass can be calculated using the formula KE = 1/2mv^2, where KE is the kinetic energy, m is the mass of the object, and v is the final velocity. This formula takes into account the mass and velocity of the object.

## 3. What factors affect the minimum energy to accelerate a mass?

The minimum energy to accelerate a mass is affected by the mass of the object, the velocity it needs to reach, and any external forces acting on the object, such as friction or air resistance. In addition, the type of energy used to accelerate the mass, such as mechanical or electrical energy, can also impact the minimum energy required.

## 4. Can the minimum energy to accelerate a mass be decreased?

Yes, the minimum energy to accelerate a mass can be decreased by reducing the mass of the object or by reducing the final velocity required. For example, using a smaller and lighter object or increasing the time taken to reach the desired velocity can decrease the minimum energy needed.

## 5. Why is understanding the minimum energy to accelerate a mass important?

Understanding the minimum energy to accelerate a mass is important in various fields of science and engineering, such as transportation, mechanics, and energy production. It helps in designing efficient and cost-effective systems and optimizing the use of energy in various processes.

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