To determine the work required to accelerate a proton from rest to a given kinetic energy, we need to use the work-energy theorem from physics. This states that the net work done on an object equals its change in kinetic energy.
The kinetic energy of an object is given by:
KE = 1/2 mv^2
Where m is the mass of the object and v is its velocity. For a proton, the mass is extremely small, around 1.67×10^-27 kg.
So if we know the proton’s mass and its final kinetic energy, we can use the work-energy theorem to calculate the net work required to accelerate it from rest (initial velocity = 0) to that final kinetic energy and velocity. The initial kinetic energy at rest is zero.
Example Calculation
Let’s look at a sample problem to illustrate how this calculation is done:
Suppose we want to know the work required to accelerate a proton from rest to a kinetic energy of 10 MeV (mega-electron volts).
1 MeV = 1.6 x 10^-13 Joules (conversion factor between MeV and Joules)
So 10 MeV = 10 x 1.6 x 10^-13 J = 1.6 x 10^-12 J
This is the final kinetic energy of the proton.
To find the work required, we use the work-energy theorem:
Wnet = ΔKE
Wnet is the net work done on the proton.
ΔKE is the change in kinetic energy of the proton.
We know:
Initial KE = 0 (at rest)
Final KE = 1.6 x 10^-12 J (from the problem statement)
Therefore:
ΔKE = Final KE – Initial KE
= 1.6 x 10^-12 J – 0
= 1.6 x 10^-12 J
Plugging this into the work-energy theorem:
Wnet = ΔKE
= 1.6 x 10^-12 J
Therefore, the net work required to accelerate this proton from rest to 10 MeV of kinetic energy is 1.6 x 10^-12 J.
Relation to Velocity
We can also relate the kinetic energy to the final velocity of the proton using:
KE = 1/2 mv^2
Rearranging this:
v = sqrt(2KE/m)
Where v is the final velocity, KE is the kinetic energy, and m is the mass of the proton.
Plugging in the numbers from our example:
KE = 1.6 x 10^-12 J
m = 1.67 x 10^-27 kg
v = sqrt(2 x 1.6 x 10^-12 J / 1.67 x 10^-27 kg)
v = 1.58 x 107 m/s
So for a final kinetic energy of 10 MeV, the proton’s final velocity is 1.58 x 107 m/s.
Trends
Some trends we can notice from these calculations:
- The lower the final kinetic energy, the less work is required to accelerate the proton from rest.
- The proton’s final velocity increases with the square root of the final kinetic energy.
- Because the proton mass is so small, even a relatively modest amount of kinetic energy requires an enormous velocity.
These trends apply for any particle being accelerated – the lighter the particle, the more energy and velocity it gains for a given amount of work.
Kinetic Energy Ranges
Protons can be accelerated to a wide range of kinetic energies for various applications:
| Kinetic Energy Range | Application |
|---|---|
| 1-10 MeV | Medical applications like radiotherapy |
| 10-100 MeV | Production of short-lived radioisotopes for imaging |
| 100-250 MeV | Injectors for high-energy physics experiments |
| 1-7 GeV | Particle colliders like the Large Hadron Collider |
Higher energy protons require more elaborate and larger accelerators. The work required spans many orders of magnitude depending on the desired energy level.
Work Required at Extreme Energies
At the highest energies, like those achieved in the Large Hadron Collider, truly enormous amounts of work are required to accelerate protons from rest. Some example calculations:
- To accelerate a proton to 6.5 TeV (6.5 x 10^12 eV):
- KE = 6.5 x 10^12 x 1.6 x 10^-13 J = 1.04 x 10^0 J
- ΔKE = KE (since starting from rest) = 1.04 x 10^0 J
- Therefore, Wnet = 1.04 x 10^0 J
- To accelerate a proton to 7 TeV (7 x 10^12 eV):
- KE = 7 x 10^12 x 1.6 x 10^-13 J = 1.12 x 10^0 J
- ΔKE = 1.12 x 10^0 J
- Therefore, Wnet = 1.12 x 10^0 J
As you can see, accelerating protons to the highest energies requires work on the order of 10^0 Joules, which is a huge amount of work!
Conclusion
In summary, the work required to accelerate a proton from rest to a given kinetic energy can be calculated using the work-energy theorem. The proton’s mass and the desired final kinetic energy determine how much work must be done on the proton to reach that energy. In accelerator physics, a wide range of proton kinetic energies are used, each requiring successively larger amounts of work to achieve. The highest energy accelerators like the LHC require truly massive amounts of work to accelerate protons to energies in the trillions of electron volts.