Since a microwave is a part of the electromagnetic spectrum, its velocity will be equal to the speed of light:

$\displaystyle 3*10^8\frac{m}{s}$

Any wave on the electromagnetic spectrum will have the same velocity, though their wavelengths and frequencies will vary, defining their individual characteristics.

Both of these waves are on the electromagnetic spectrum. Any wave on the spectrum will have the same speed, the speed of light.

$\displaystyle c=3*10^8\frac{m}{s}$

Radio waves, microwaves, infrared waves, visible light, ultraviolet light, and gamma rays are all on the spectrum. They all have the same ultimate speed, but vary in their wavelengths, frequencies, and energy levels.

The relationship between wavelength, frequency, and velocity is:

$\displaystyle v=f\lambda$

In this case, because infrared light is a part of the electromagnetic spectrum, we would use $\displaystyle c$ for $\displaystyle v$.

$\displaystyle c=f\lambda$

We are given the speed of light and the frequency of the photon. Using these values, we can solve for the wavelength.

$\displaystyle 3*10^8\frac{m}{s}=(2.5*10^{14}Hz)\lambda$

$\displaystyle \lambda=\frac{3*10^8\frac{m}{s}}{2.5*10^{14}Hz}$

$\displaystyle \lambda=1.2*10^{-6}m$

The relationship between velocity, frequency, and wavelength of a waveform is given by the formula:

$\displaystyle v=f\lambda\ \text{or}\ f = \frac{v}{\lambda}$

In this case, we are dealing with an electromagnetic wave. All electromagnetic waves travel at the same velocity: the speed of light.

Using this value and the given frequency of the wave, we can solve for the wavelength.

$\displaystyle 3.0*10^8\frac{m}{s}=(4*10^{14}Hz)\lambda$

$\displaystyle \lambda=\frac{3.0*10^8\frac{m}{s}}{4*10^{14}Hz}=7.5*10^{-7}m$

Wavelength is traditionally reported in nanometers, corresponding to $\displaystyle \small 10^{-9}$.

$\displaystyle 7.5*10^{-7}m=750*10^{-9}m=750nm$

Any wave on the electromagnetic spectrum will travel at the same constant speed, the speed of light. Each type of wave is determined by changes in the frequency, wavelength, and energy, but these factors always reflect a constant velocity. The types of waves in the electromagnetic spectrum are radio waves, microwaves, infrared light, visible light, ultraviolet light, X-rays, and gamma rays. Radio waves will have the largest wavelength, smallest frequency, and lowest energy. Gamma rays will have the smallest wavelength, highest frequency, and greatest energy.

The inverse relationship between frequency and wavelength means that the velocity can remain constant.

Photons can move at the speed of light because they have no mass. Anything with mass can not be accelerated to the speed of light—it takes an infinite amount of energy to do so. We can get very close, but never exactly there.

For this problem, use Snell's law: $\displaystyle n_1\sin(\Theta_1)=n_2\sin(\Theta_2)$.

In this equation, $\displaystyle \small n$ is the index of refraction and $\displaystyle \Theta$ is the angle of refraction to the vertical. Using the values given in the question, we can find the resultant angle of refraction.

$\displaystyle n_1\sin(\Theta_1)=n_2\sin(\Theta_2)$

$\displaystyle 1.00\sin(30^{\circ})=1.33\sin(\Theta_2)$

$\displaystyle 0.5=1.33\sin(\Theta_2)$

$\displaystyle \frac{0.5}{1.33}=\sin(\Theta_2)$

$\displaystyle 0.376=\sin(\Theta_2)$

Take the arcsin of both sides to find the value of $\displaystyle \Theta_2$.

$\displaystyle \sin^{-1}(0.376)=\Theta_2$

$\displaystyle 22.1^{\circ}=\Theta_2$

The relationship between speed of light and the index of refraction is:

$\displaystyle n=\frac{c}{v}$

In this formula, $\displaystyle c$ is the speed of light in a vacuum, $\displaystyle v$ is the observed speed of light in the substance, and $\displaystyle n$ is the index of refraction. We are given the speed of light in a vacuum and the index of refraction, allowing us ot solve for the speed of light in the crystal.

$\displaystyle 3.33=\frac{3*10^8\frac{m}{s}}{v}$

$\displaystyle v=\frac{3*10^8\frac{m}{s}}{3.33}$

$\displaystyle v=9*10^7\frac{m}{s}$

The relationship between speed of light and the index of refraction is:

$\displaystyle n=\frac{c}{v}$

In this formula, $\displaystyle c$ is the speed of light in a vacuum, $\displaystyle v$ is the observed speed of light in the substance, and $\displaystyle n$ is the index of refraction. We given the values for the speed of light in the crystal and the speed of light in a vacuum, allowing us to solve for the index of refraction.

Plug in our given values and solve.

$\displaystyle n=\frac{3*10^8\frac{m}{s}}{2*10^8\frac{m}{s}}$

$\displaystyle n=1.5$

Note that the index of refraction is a ratio of two velocities, and therefore has no units.

The index of refraction is defined as the relationship between the speed of light in a vacuum over the speed of light in the gas/liquid/solid.

$\displaystyle n=\frac{c}{v}$

We know the index of refraction of the gas, and the speed of light in a vacuum. Using these values, we can solve for the speed of light in the gas.

$\displaystyle 1.32=\frac{3*10^8\frac{m}{s}}{v}$

$\displaystyle v=\frac{3*10^8\frac{m}{s}}{1.32}$

$\displaystyle v=2.27*10^8\frac{m}{s}$