A relativistic proton has a momentum of 1.0×10^-17 kg ∙ m/s and a rest energy of 0.15 nJ. What is the kinetic energy of
this proton? (c = 3.00 × 108 m/s, mass of proton = 1.67 × 10-27 kg)
Well, let's not get too caught up in the science jargon. The kinetic energy of a proton is like the energy it has when it's on the move, ready to cause some mischief!
To calculate the kinetic energy, we need to know the mass of the proton and its velocity. Given that the mass of a proton is 1.67 × 10^-27 kg, we got that covered.
Now, let's put on our superhero capes and calculate the velocity of this speedy proton. We know its momentum is 1.0 × 10^-17 kg ∙ m/s. But we also know the formula for momentum is p = mv, where p is the momentum, m is the mass, and v is the velocity.
So, rearranging the equation, we find that the velocity of our proton is v = p/m. Plugging in the values, we get v = 1.0 × 10^-17 kg ∙ m/s / (1.67 × 10^-27 kg).
Calculating that, we get an astounding velocity of about 5.988 × 10^9 m/s. That proton must have had a looooooot of coffee!
Now, the kinetic energy of our speedy proton can be calculated using the formula KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
Let's plug in our values: KE = (1/2)(1.67 × 10^-27 kg)(5.988 × 10^9 m/s)^2.
After crunching the numbers, we find that the kinetic energy of this speedy proton is approximately 2.391 × 10^-8 Joules, or in clown units, about enough energy to power a hilariously bad joke for at least a month!
To find the kinetic energy of the proton, we need to calculate its relativistic mass using the formula:
m = m0 / sqrt(1 - v^2/c^2)
where:
m = relativistic mass
m0 = rest mass
v = velocity of the proton
c = speed of light
First, let's calculate the velocity of the proton using its momentum:
p = m * v
v = p / m
Given:
Momentum (p) = 1.0 × 10^(-17) kg * m/s
Rest mass (m0) = 1.67 × 10^(-27) kg
Speed of light (c) = 3.00 × 10^8 m/s
v = p / m
v = 1.0 × 10^(-17) kg * m/s / 1.67 × 10^(-27) kg
v = 5.99 × 10^9 m/s
Now, let's calculate the relativistic mass:
m = m0 / sqrt(1 - v^2/c^2)
m = 1.67 × 10^(-27) kg / sqrt(1 - (5.99 × 10^9 m/s)^2 / (3.00 × 10^8 m/s)^2)
m ≈ 1.88 × 10^(-27) kg
Next, we can calculate the kinetic energy using the formula:
KE = (γ - 1) * m0 * c^2
where:
KE = kinetic energy
γ = Lorentz factor (γ = 1 / sqrt(1 - v^2/c^2))
m0 = rest energy
c = speed of light
First, let's calculate the Lorentz factor:
γ = 1 / sqrt(1 - v^2/c^2)
γ = 1 / sqrt(1 - (5.99 × 10^9 m/s)^2 / (3.00 × 10^8 m/s)^2)
γ ≈ 3.47
Now, let's calculate the kinetic energy:
KE = (γ - 1) * m0 * c^2
KE = (3.47 - 1) * 0.15 nJ * c^2
KE = (2.47) * (0.15 × 10^(-9) J) * (3.00 × 10^8 m/s)^2
KE ≈ 1.11 × 10^(-15) J
Therefore, the kinetic energy of the proton is approximately 1.11 × 10^(-15) Joules.
To find the kinetic energy of a relativistic proton, we need to use the equation that relates energy and momentum in special relativity.
The equation is:
E^2 = (mc^2)^2 + (pc)^2
where E is the total energy of the proton, m is its rest mass, c is the speed of light, and p is the momentum of the proton.
Given that the momentum (p) of the proton is 1.0×10^-17 kg ∙ m/s and the rest energy (mc^2) is 0.15 nJ, we need to solve for the kinetic energy (E).
First, let's rearrange the equation to solve for E:
E = sqrt((mc^2)^2 + (pc)^2)
Now, we can substitute the given values into the equation and calculate the kinetic energy:
E = sqrt((0.15 nJ)^2 + (1.0×10^-17 kg ∙ m/s × 3.00 × 10^8 m/s)^2)
E = sqrt((0.15 × 10^-9 J)^2 + (1.0 × 10^-17 kg ∙ m/s × 3.00 × 10^8 m/s)^2)
E = sqrt((2.25 × 10^-19 J^2) + (9.0 × 10^-9 kg^2 ∙ m^2/s^2))
E = sqrt(2.25 × 10^-19 J^2 + 9.0 × 10^-9 kg^2 ∙ m^2/s^2)
E = sqrt(2.25 × 10^-19 J^2 + 9.0 × 10^-19 J^2)
E = sqrt(11.25 × 10^-19 J^2)
E = sqrt(11.25 × 10^-19) J
E = sqrt(11.25) × sqrt(10^-19) J
E = 3.35 × 10^-10 J
So, the kinetic energy of the proton is approximately 3.35 × 10^-10 Joules.