OK, so I'm having some serious problems with this one question. It's on relativity. Here it is:
Calculate the mass of a proton (m0 = 1.67E-27 kg) whose kinetic energy is one sixth its total energy. How fast is it traveling?
THANKS!!
Total energy is then
Etotal = m0c^2 + KE = (6/5) m0 c^2
= m0 c^2/sqrt[1 - (v/c)^2]
6/5 = 1/sqrt[1 - (v/c)^2]
25/36 = 1 - (v/c)^2
(v/c)^2 = 11/36
v/c = 0.306
v = 9.17*10^7 m/s
You don't need to use the rest mass.
There is a nonrelativistic way to do this also, but you will get a different (and wrong) answer. That would be to calculate m0c^2, take 1/5 of it for the kinetic energy, and set that equal to
(1/2) m0 v^2, then solve for v.
wait, aren't we lookng for the mass, too?
Whoops, you are right. They asked for mass, too. It is the rest mass times the factor 1/sqrt[1 - (v/c)^2], which is (6/5) m0.
m = m0 / √ [1 - v^2/c^2]
Kinetic energy = [m-m0] c^2.
Total energy = mc^2.
It is given that [m-m0] c^2 = mc^2/6
m = [6/5] *m0 = [6*1.67e-27/5] = 2.004e-27 kg.
--------------------------------------…
v^2 /C^2=[ m^2 -m0^2] / m^2 =[ 2.004e-27 ^2 -1.67e-27^2] /2.004e-27^2
v = 0.553 C where C = 3e8m/s.
=================================
what would happen if it became 1/4 the total energy instead? how would the equation be set up?
To solve this problem, we need to apply the principles of relativity. The equation we will be using is the energy-mass equivalence equation:
E = mc^2
where E is the total energy, m is the mass, and c is the speed of light.
Given that the proton's kinetic energy (KE) is one-sixth of its total energy, we can express the total energy as:
E = KE + rest energy
Since KE = (1/6)E, the rest energy is given by:
rest energy = E - KE
= E - (1/6)E
= (5/6)E
To find the mass, we can rearrange the energy-mass equivalence equation:
E = mc^2
m = E / c^2
Therefore, the rest mass m0 is given by:
m0 = rest energy / c^2
Substituting the value of rest energy, we get:
m0 = (5/6)E / c^2
Given that m0 = 1.67E-27 kg, we can now calculate the mass m of the proton.
To find the velocity at which the proton is traveling, we need to use the formula for kinetic energy:
KE = (1/2)mv^2
Rearranging this equation gives us:
v^2 = (2KE) / m
v = sqrt((2KE) / m)
Now that we know the kinetic energy (KE) and mass (m), we can substitute those values into the equation to find the velocity (v) at which the proton is traveling.
I hope this helps you understand how to approach the problem.