Video Transcript
Find the momentum of a helium
nucleus having a mass of 6.68 times 10 to the negative 27th kilograms that is moving
at 0.200𝑐.
We can call the given mass, 6.68
times 10 to the negative 27th kilograms, 𝑚. And we can call the nucleus’ speed,
0.200𝑐, 𝑣. We want to solve for the nucleus’s
momentum. We’ll call that 𝑝. Now, starting off, we may not know
at first whether to use the classical or the relativistic formulation for
momentum. The clue in the problem statement
that tells us to use relativistic is the fact that the speed of this nucleus is
comparable to the speed of light. That means relativistic effects
will be important to include.
Relativistic momentum 𝑝 is equal
to 𝛾 times rest mass, 𝑚 sub zero, times 𝑣, where 𝛾 is one over the square root
of one minus 𝑣 squared over 𝑐 squared. So, 𝑝 equals 𝑚𝑣 over the square
root of one minus 𝑣 squared over 𝑐 squared, where 𝑣 is given to us in the problem
statement as two-tenths the speed of light 𝑐. And 𝑚 also is given to us in the
statement. So, we’re now ready to plug in to
solve for momentum.
When we enter our values to this
equation, the factor of the speed of light 𝑐 under the square root sign cancels
out. But in the numerator, it
remains. In this exercise, we’ll treat 𝑐 as
exactly 3.00 times 10 to the eighth meters per second. When we make that substitution and
then enter these values into our calculator, we find that 𝑝 is 4.09 times 10 to the
negative 19th kilograms meters per second. That’s the relativistic momentum of
this helium nucleus.
