Video: Calculating the Speed of a Proton Corresponding to a Certain de Broglie Wavelength

A proton has a rest mass of 1.67 × 10⁻²⁷ kg. At what speed would a proton have to move in order to have a de Broglie wavelength of 8.82 × 10⁻⁹ m? Use a value of 6.63 × 10⁻³⁴ J⋅s for the Planck constant. Give your answer to 3 significant figures.

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Video Transcript

A proton has a rest mass of 1.67 times 10 to the negative 27th kilograms. At what speed would a proton have to move in order to have a de Broglie wavelength of 8.82 times 10 to the negative ninth meters? Use a value of 6.63 times 10 to the negative 34th joule-seconds for the Planck constant. Give your answer to three significant figures.

Okay, so in this example, we have a proton. And we’re told that this proton has a rest mass, we’ll call it 𝑚 sub zero, of 1.67 times 10 to the negative 27th kilograms. Now, an object’s rest mass is the mass it possesses when it’s being measured by an observer at rest relative to the object. That is, compared to the observer, the object is not in motion. What we want to consider is how fast does a proton with this rest mass have to move in order to have this specific de Broglie wavelength, 8.82 times 10 to the negative ninth meters.

So let’s say that we want to solve for the velocity 𝑣 of this proton. And in support of finding this, we can say that the given de Broglie wavelength of the proton is represented by 𝜆 sub B. At this point, we can recall that the de Broglie wavelength of any object is equal to Planck’s constant ℎ divided by the momentum of that object. In other words, it’s equal to ℎ divided by 𝑚 times 𝑣, the object’s mass and its velocity.

Now, in our case, it’s not the de Broglie wavelength of the proton we want to solve for, but rather its speed. So let’s rearrange this equation to solve for 𝑣. To do this, let’s multiply both sides by 𝑣 divided by 𝜆 B. This cancels out the de Broglie wavelength on the left. And it cancels out the speed 𝑣 on the right. So then, the proton speed we want to solve for is equal to Planck’s constant divided by the mass of the proton times its de Broglie wavelength. Regarding these values, we’re to use a value of 6.63 times 10 to the negative 34th joule-seconds for ℎ. We’re given the de Broglie wavelength 𝜆 B of the proton. And for its mass, we’ll use its rest mass, 𝑚 sub zero, given as 1.67 times 10 to the negative 27th kilograms.

With all these values plugged in, note that all three are given to a precision of three significant figures. Our final answer then will have that same number. Calculating this result, we find an answer of 45.0 meters per second. This is a fairly low speed, certainly nonrelativistic. And it’s the speed the proton would need to have in order to have the given de Broglie wavelength.

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