Energy Needed To Go A Given Speed

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To determine the energy required to accelerate a mass to a given speed, the kinetic energy formula changes based on the speed relative to the speed of light. For low speeds, the equation is KE = ½mv², while for speeds approaching light speed, it becomes mc²/√(1 - v²/c²). The additional energy needed to reach speed v is calculated as mc²(1/√(1 - v²/c²) - 1). For example, accelerating a 20-kilogram mass to 0.1c requires approximately 9e+12 kilojoules. Understanding these equations is crucial for calculations involving relativistic speeds.
Sothh
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Hey, I don't have much of a physics background.

I have been searching for an equation that shows how much energy is needed to make a given mass go a given speed.

For example, let's say I want to send 1 gram at half light speed. How much energy is required?

Thanks!
 
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Sounds like you want to calculate the kinetic energy of the mass.

For low speeds (compared to the speed of light) the kinetic energy is KE = ½mv². But for fast speeds, it's a bit more complicated:
76cc95f8c7e7d0edde43428308072182.png


(I can't seem to get Latex to work.)
 
Welcome to PF!

Hi Sothh! Welcome to PF! :smile:

For a speed of v metres per second, the energy of a mass of m kilograms is mc2/√(1 - v2/c2) joules.

If v = 0 (the mass is at rest), that energy is obviously just mc2, which is therefore called the rest-mass.

So the extra energy needed to get it to speed v (the kinetic energy) is mc2(1/√(1 - v2/c2) - 1) joules. :wink:

If v/c is very small, then that extra (kinetic) energy is approximately mc2(v2/2c2), ie mv2/2.
 
Thanks!

Just to confirm I am using this correctly, it should take 20 kilograms around 9e+12 kilojoules to go .1c, correct?
 
Sothh said:
Just to confirm I am using this correctly, it should take 20 kilograms around 9e+12 kilojoules to go .1c, correct?
Sounds about right.
 

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