A particle of mass m0 is given a kinetic energy equal to one-third its rest-mass energy. How fast must the particle be traveling? please include units
Kinetic energy =
Total energy - m0*c^2
= (1/3)*m0*c^2
Total energy = (4/3)m0*c^2
= m0*c^2/sqrt[1 - (v/c)^2]
sqrt[1 - (v/c)^2] = 3/4
1 - (v/c)^2 = 9/16
(v/c)^2 = 7/16
v/c = 0.661
You don't need units in that dimensionless form.
To find the speed of the particle, we need to make use of the formula for the kinetic energy of a moving object:
Kinetic energy (K.E.) = (1/2) * m * v^2
where m is the mass of the object and v is its velocity.
The rest-mass energy of the particle (E) is given by Einstein's famous equation:
E = m0 * c^2
where m0 is the rest mass of the particle and c is the speed of light.
In the given problem, the kinetic energy of the particle is one-third of its rest-mass energy. So we have:
K.E. = (1/3) * E
Substituting the equations for K.E. and E, we get:
(1/2) * m * v^2 = (1/3) * m0 * c^2
Now, we can simplify and solve for the velocity (v):
v^2 = (2/3) * (m0 * c^2) / m
v = √((2/3) * (m0 * c^2) / m)
The units of velocity will depend on the units used for mass (m), rest mass (m0), and the speed of light (c). Usually, mass is measured in kilograms (kg) and the speed of light is approximately 3 * 10^8 meters per second (m/s).
So, the units for velocity will be meters per second (m/s).
Please note that this equation assumes non-relativistic speeds. If the speed of the particle approaches the speed of light, relativistic effects need to be considered.