### Question

Why is it easier to accelerate an electron to a speed that is close to the speed of light, compared to accelerating a proton to the same speed?

Asked by: Saira Haider

### Answer

The one-line answer is that a proton is more massive than an electron.
It's the same reason why it is more difficult (takes more gasoline) to accelerate a Ford
Excursion to 60 mph than it is to accelerate a Geo Prism to the same speed, ignoring air
friction. Using examples that travel at low, non-relativistic speeds, the explanation
becomes a little more intuitive. Let's say the Ford Excursion has a mass 'm'. Then, to
speed it up to a speed 'v', we have to add kinetic energy to the Excursion by burning
gasoline in the engine. The amount of energy we have to add is E = 0.5*m*v^{2}. Let's say
the mass of the Prism is m_{1} = 0.25*m, the Excursion weighs four times as much as the Prism.
How much energy does it take to get the Prism up to the same speed 'v' as the Excursion? We
just use the same formula, but use 'm_{1}' for the mass instead of 'm': E = 0.5*m_{1}*v^{2}. But,
we know how m_{1} and m are related, so we can use this knowledge: E = 0.5*m_{1}*v^{2} =
0.5*(0.25*m)*v^{2} = 0.125*m*v^{2}. In plain English, it takes four times as much energy to
accelerate a Ford Excursion up to a speed 'v'Keep in mind that I'm ignoring air drag, tire
friction, and engine efficiency in this simple example. These have in important effect on
the actual fuel economy of cars, but they don't come into play when we are talking about
electrons and protons.

When we get up to speeds approaching the speed of light, the formulas relating speed and
energy input look different, but the answer is the same: The proton's mass is larger, so it
needs more energy to achieve a certain speed than the electron.

Answered by: Andreas Birkedal-Hansen, M.A., Physics Grad Student, UC Berkeley

The short answer is pretty simple: Because the mass
of the proton is about 1840 times of that of an electron.
If we think just in non-relativistic terms, applying the
same force to the two objects, the electron will accelerate
much more, since F=ma, as said by Newton, the greater the
mass, the less will the acceleration be.

However, the question at hand in a way defies non-relativistic
handling, because what is asked is 'a speed close to the
speed of light'. In that case, the concept of force is to
some extent on shaky ground, but we can do better just
considering the amount of energy we must give the particle
to accelerate it to a certain speed, and find the relation
between speed, energy and mass, and the answer will be
obvious.

The relativistic energy of a particle is

E = mc^{2} / sqrt(1-v^{2}/c^{2})

Where m is its (rest) mass, c is the speed of light, v is
the speed of the particle. For v << c (slow speeds)
it reduces to :

E = mc^{2} + 1/2 m*v^{2}

Here, mc^{2} is the 'rest energy'.

Now, at a speed v, the Kinetic energy is therefore

T = mc^{2} (1/sqrt(1-v^{2}/c^{2}) - 1)

Which is the energy you have to give the particle to make
it travel at the speed v. It is obviously linear with
m -- the bigger the mass, the greater energy you need,
pretty much the same case as non-relativistic, as expected.

Answered by: Yasar Safkan, B.S., Physics Ph.D. candidate, MIT