Question

What is the frequency of a photon that has the same momentum as a neutron moving with a speed of 1.90 × 103 m/s?

Answer 1

The mass of a neutron is:
$m=1.67 \cdot 10^{-27}kg$
Since we know its speed, we can calculate the neutron's momentum:
$p=mv=(1.67 \cdot 10^{-27}kg)(1.90 \cdot 10^3 m/s)=3.17 \cdot 10^{-24} kg m/s$

The problem says the photon has the same momentum of the neutron, p.  The photon momentum is given by
$p= \frac{h}{\lambda}$
where h is the Planck constant, and $\lambda$ is the photon wavelength. If we re-arrange the equation and we use the momentum we found before, we can calculate the photon's wavelength:
$\lambda= \frac{h}{p}= \frac{6.6 \cdot 10^{-34}Js}{3.17 \cdot 10^{-24} kg m/s}=2.08 \cdot 10^{-10} m$

And since we know the photon travels at speed of light c, we can now calculate the photon frequency:
$f= \frac{c}{\lambda}= \frac{3 \cdot 10^8 m/s}{2.08 \cdot 10^{-10} m}= 1.44 \cdot 10^{18} Hz$