A proton moving with a constant speed has a total energy 2.5 times its rest energy. What is the proton's:
a. speed
b. kinetic energy
Kinetic energy is the energy minus the rest energy, so that is 1.5 times the rest energy.
The speed follows from E = gamma m c^2 = gamma times the rest energy.
So, gamma = 2.5 ------->
v = 0.917 c
To find the answers, we need to use the equation for total relativistic energy:
E = γmc²
where:
E = total energy
γ = Lorentz factor (γ = 1/√(1 - v²/c²))
m = rest mass of the proton
c = speed of light in a vacuum
Let's denote the total energy as E_total and the rest energy as E_rest. We are given that:
E_total = 2.5E_rest
To find the speed of the proton, we can rearrange the equation E_total = γmc² as follows:
γ = E_total / (mc²)
Now, γ can be expressed as:
γ = 1/√(1 - v²/c²)
Squaring both sides and rearranging, we get:
(1 - v²/c²) = 1 / γ²
Substituting for γ:
(1 - v²/c²) = mc² / E_total²
Now, we can solve for v²/c²:
v²/c² = 1 - (mc² / E_total²)
Substituting the given values:
v²/c² = 1 - (m / (2.5E_rest))²
a. To find the speed, we need to calculate v/c by taking the square root of the above expression:
v/c = √(1 - (m / (2.5E_rest))²)
b. The kinetic energy can be found using the equation for kinetic energy:
K = (γ - 1)mc²
Substituting the value of γ from above:
K = (E_total / (mc²)) - 1)mc²
= E_total - mc²
Now, we can substitute the given values to find the kinetic energy:
K = 2.5E_rest - m × c²
Please provide the rest energy of the proton so we can calculate the values accurately.
To solve this problem, we need to use the concept of relativistic energy and the equation for total energy.
a. To find the speed (v) of the proton, we can use the equation for total energy (E):
E = γmc²,
where E is the total energy of the proton, γ is the Lorentz factor, m is the rest mass of the proton, and c is the speed of light.
Given that the total energy (E) is 2.5 times the rest energy (m₀c²), we can write the equation as:
E = 2.5m₀c².
Since γ = E / (m₀c²), we have:
γ = 2.5.
To find the speed (v), we can use the equation:
γ = 1 / sqrt(1 - (v² / c²)).
Squaring both sides of the equation, we get:
γ² = 1 - (v² / c²).
Rearranging the equation, we have:
(v² / c²) = 1 - (1 / γ)².
Substituting γ = 2.5, we get:
(v² / c²) = 1 - (1 / 2.5)².
Simplifying further:
(v² / c²) = 1 - (1 / 6.25).
(v² / c²) = 1 - 0.16.
(v² / c²) = 0.84.
To find v, we take the square root of both sides:
sqrt(v² / c²) = sqrt(0.84).
v / c = 0.917.
Multiplying both sides by c, we get:
v = c * 0.917.
Therefore, the speed of the proton is approximately 0.917 times the speed of light (c).
b. To find the kinetic energy, we use the equation for kinetic energy (K):
K = (γ - 1)mc².
Given that γ = 2.5 and m is the rest mass of the proton, we can directly calculate the kinetic energy using the above equation:
K = (2.5 - 1)mc².
K = 1.5mc².
Therefore, the kinetic energy of the proton is 1.5 times its rest energy (m₀c²).