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Special relativity: find speed and kinetic energy

  1. Apr 28, 2010 #1
    1. The problem statement, all variables and given/known data

    A proton (rest mass 1.67x10^-27 kg) has total energy that is 3.2 times its rest energy. What is:
    a) the kinetic energy of the proton (in joules)
    b) the magnitude of the momentum of the proton (in kg*m/s)
    c) the speed of the proton (in terms of the speed of light "c")


    2. Relevant equations

    E(0) = m(0)c^2
    where E(0) is rest energy, m(0) is rest mass and c is the speed of light (approximately 3.00x10^8 m/s)

    E = mc^2
    where E is the total energy (im not too sure if thats the formula for total energy) and m is the relativistic mass

    p=mv
    where p is the momentum

    E^2 = (p^2)(c^2) + (m(0)^2)(c^2)

    Ke = E - m(0)c^2
    where Ke is the kinetic energy

    3. The attempt at a solution

    I found the rest energy E(0) = m(0)c^2 = 1.503x10^-10 and since my total energy = 3.2 times the rest energy, E = 3.2E(0) = 4.81x10^-10.
    And because E = mc^2, i can use the previously calculated value to find m which gave me m= 5.344x10^-27.
    I used these values and the equation E^2 = (p^2)(c^2) + (m(0)^2)(c^2)to find the momentum which gave me 1.603x10^-18, which is correct.

    But when i use the formula p = mv and rearrange it as v = p/m, i get 0.9999c m/s which according to my homework isn't correct.

    And also, when i use the equation Ke = E - m(0)c^2 to find the kinetic energy, i get 3.307x10^-10, which is also incorrect.

    I know it's long but a little help would be appreciated.
     
  2. jcsd
  3. Apr 28, 2010 #2
    Relativistic momentum is [tex]p=\gamma m_0v[/tex] and relativistic kinetic energy is [tex](\gamma -1)m_0c^2[/tex] with [tex]\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}[/tex]

    See if you get the right result with these expressions.
     
  4. Apr 28, 2010 #3
    but how would i calculate the gamma value if i don't have the velocity?
     
  5. Apr 28, 2010 #4
    Total energy is [tex]E=\gamma m_0c^2[/tex] and rest energy is [tex]E_0=m_0c^2[/tex]. Now take another look at the problem statement. Can you figure out what the velocity and the value of gamma is?
     
  6. Apr 28, 2010 #5
    Ok then suppose that before i look for the velocity, i want to find the kinetic energy first. If i use the equations E= (gamma)m(0)c^2 and E(0) = m(0)c^2 , i can find the value for gamma, which gives me 3.2. if i then use the equation (gamma -1)m(0)c^2 to find the kinetic erergy it gives me 3.307x10^-10. Why is this answer wrong?
     
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