Calculate the speed of a
1.00-microgram-mass dust particle that has the same momentum as a proton moving at
0.999𝑐. The rest mass of a proton is 1.67
times 10 to the negative 27th kilograms.
We can call the 1.00-microgram mass
of the dust particle 𝑚 sub 𝑑. And we can call the proton’s mass
of 1.67 times 10 to the negative 27th kilograms 𝑚 sub 𝑝. The proton’s speed, 0.999𝑐, we’ll
call 𝑣 sub 𝑝. We want to solve for the speed of
the dust particle, which we’ll call 𝑣 sub 𝑑.
We can start off our solution by
calculating the relativistic momentum of the proton. We recall the relationship for
relativistic momentum, that 𝑝 is equal to rest mass 𝑚 sub zero times 𝑣 over the
square root of one minus 𝑣 squared over 𝑐 squared. If we call the relativistic
momentum of our proton 𝑝 sub 𝑝, then that’s equal to 𝑚 sub 𝑝 𝑣 sub 𝑝 over the
square root of one minus 𝑣 sub 𝑝 squared over 𝑐 squared. In the problem statement, we’re
told 𝑚 sub 𝑝 and 𝑣 sub 𝑝, so we can plug those into this equation now.
When we do, when we look under the
square root sign, we see that the factors of 𝑐 cancel one another out. But the factor of 𝑐 in the
numerator remains. We’ll treat the speed of light 𝑐
as exactly 3.00 times 10 to the eighth meters per second. Using that value and entering these
terms on our calculator to solve for 𝑝 sub 𝑝, result is 1.119 times 10 to the
negative 17th kilograms meters per second. We’ve now solved for 𝑝 sub 𝑝, the
momentum of the proton, but what we’re after is 𝑣 sub 𝑑, the speed of the dust
If we call 𝑝 sub 𝑑 the momentum
of the dust particle, we know that that’s equal to 𝑝 sub 𝑝. But we don’t yet know whether we’ll
need to include relativistic effects to solve for it.
If we rewrite the mass of the dust
particle into units of kilograms from units of micrograms, then it’s equal to
[1.00] times 10 to the negative ninth kilograms.
If we took the ratio of the mass of
the dust particle to the mass of the proton, the dust particle is roughly 10 to the
18th times more massive than the proton. Which means that the speed of the
dust particle, 𝑣 sub 𝑑, related to the speed of the proton is the inverse of that,
roughly 10 to the negative 18th times as fast.
This tells us that 𝑣 sub 𝑑 is
slow enough that we may reasonably neglect relativistic effects when we calculate
it. That means we can replace 𝑝 sub 𝑑
with 𝑚 sub 𝑑 times 𝑣 sub 𝑑. Solving for 𝑣 sub 𝑑, it’s equal
to the proton’s momentum divided by the dust particle’s mass. These values we know and can plug
in. When we calculate this fraction, we
find that 𝑣 sub 𝑑, the speed of the dust particle, is 1.12 times 10 to the
negative eighth meters per second. That’s how fast the dust particle