Estimate the De Broglie wavelength of:

a) The Moon orbiting the Earth;

b) A nitrogen molecule at room temperature;

c) A well-served tennis ball;

d) An alpha particle emitted in the radioactive decay of 210Po;

e) A beta particle emitted in the radioactive decay of 14C (the isotope used for radiocarbon dating);

f) A thermal neutron in a nuclear reactor;

g) A human skydiver at their terminal velocity

To estimate the De Broglie wavelength, we will use the formula:

λ = h / p

where λ is the De Broglie wavelength, h is the Planck's constant, and p is the momentum of the object.

a) For the Moon orbiting the Earth:
To estimate the De Broglie wavelength of the Moon, we need the mass and velocity of the Moon. However, since the Moon's mass and velocity are both quite large, the resulting De Broglie wavelength will be extremely small and practically negligible.

b) For a nitrogen molecule at room temperature:
The mass of a nitrogen molecule (N2) is approximately 28 atomic mass units (amu). We can use the average velocity of gas molecules at room temperature, which is around 500 m/s. Now, we can calculate the momentum:

p = m * v
= (28 amu) * (500 m/s)

To convert the mass to kilograms, we can use the conversion factor: 1 amu = 1.66054 x 10^-27 kg. This gives us:

m = 28 amu * (1.66054 x 10^-27 kg / 1 amu)
≈ 4.644 x 10^-26 kg

Now, let's calculate the momentum:

p = (4.644 x 10^-26 kg) * (500 m/s)

Finally, we can calculate the De Broglie wavelength using the formula:

λ = h / p

Substituting the values of Planck's constant h (approximately 6.626 x 10^-34 J·s) and momentum p, we can find the estimated De Broglie wavelength.

c) For a well-served tennis ball:
To estimate the De Broglie wavelength of a tennis ball, we need to know its mass and velocity. A standard tennis ball has a mass of approximately 58 grams (0.058 kg) and an average velocity of around 40 m/s. We can calculate the momentum and then use it in the De Broglie wavelength formula.

d) For an alpha particle emitted in the radioactive decay of 210Po:
An alpha particle is composed of two protons and two neutrons, giving it a mass of approximately 4 atomic mass units (amu). Its typical velocity can be around 2.5 x 10^7 m/s. Similar to the nitrogen molecule example, we can calculate the momentum and then determine the De Broglie wavelength.

e) For a beta particle emitted in the radioactive decay of 14C:
A beta particle is an electron or positron emitted during beta decay. For the De Broglie wavelength estimation, we need the mass and velocity of the beta particle. The mass of an electron is approximately 9.10938356 x 10^-31 kg, and its typical velocity can be around 2.2 x 10^6 m/s. As with the previous examples, we can calculate the momentum and obtain the De Broglie wavelength.

f) For a thermal neutron in a nuclear reactor:
To estimate the De Broglie wavelength of a thermal neutron, we need to know its mass and velocity. The mass of a neutron is approximately 1.67493 x 10^-27 kg. The average velocity of thermal neutrons can be around 2,200 m/s. We will calculate the momentum and use it in the De Broglie wavelength formula.

g) For a human skydiver at their terminal velocity:
Estimating the De Broglie wavelength for a human skydiver at their terminal velocity involves understanding that the De Broglie wavelength equation applies to matter particles. Human-scale objects have incredibly small De Broglie wavelengths, making them practically negligible in this context. The De Broglie wavelength is only noticeable for objects with very small masses, high velocities, or low temperatures. For a human skydiver, their mass is large, and the terminal velocity is relatively low, resulting in an insignificant De Broglie wavelength.