An incident X-ray photon of wavelength 0.2800nm is scattered from an electron that is initially at rest. The photon is scattered at an angle of 180.0degrees and has a wavelength of 0.2849nm. Use the conservation of linear momentum to find the momentum gained by the electron.

I need help with this problem!Thanks a lot!

Sure, I'd be happy to help you with this physics problem! But before we get started, let's take a moment to appreciate the situation. I mean, can you imagine being an electron and suddenly getting hit by a photon? Talk about a shocking encounter! Good thing electrons are energetic little particles that can handle a bit of momentum exchange.

Now, let's get down to business. In this problem, we can use the conservation of linear momentum to find out how much momentum the electron gained during the scattering process. The key idea here is that momentum is conserved before and after the interaction.

The formula for linear momentum is given by p = mv, where p is momentum, m is mass, and v is velocity. Since the electron is initially at rest, its initial momentum (p_initial) is zero.

After the scattering, the photon is deflected at an angle of 180 degrees (or pi radians). This indicates a complete reversal in its direction. So, we can say that the final momentum of the photon (p_photon_final) is equal in magnitude but opposite in direction to its initial momentum.

Now, according to the conservation of linear momentum, the total initial momentum must equal the total final momentum. In other words, p_initial = p_photon_final + p_electron_final.

Since the electron is initially at rest, its final momentum (p_electron_final) is equal to its kinetic momentum. We'll call the mass of the electron m_electron and its final velocity v_electron_final.

Since the photon is massless, the formula for its momentum is p_photon = hf/c, where h is Planck's constant, f is frequency, and c is the speed of light. We can relate the initial and final photon momenta by the equation p_photon_initial = -p_photon_final, since they're equal in magnitude but opposite in direction.

Since we know the wavelengths of the incident and scattered photons, we can relate their momenta using p_photon_initial = h/lambda_initial and p_photon_final = h/lambda_final, where lambda is wavelength.

Putting it all together, we have:
0 = -p_photon_final + (m_electron * v_electron_final)

Now, let's substitute the expressions for p_photon_final, p_photon_initial, and their relationship:
0 = -(h/lambda_final) + (h/lambda_initial) + (m_electron * v_electron_final)

After some rearranging, we get:
m_electron * v_electron_final = (h/lambda_final) - (h/lambda_initial)

Now, we have all the values we need to calculate the momentum gained by the electron. Plug in the respective values for Planck's constant (h), the wavelengths (lambda_initial and lambda_final), and solve for m_electron * v_electron_final.

Remember, my friend, the key to physics problem-solving is to keep calm and carry on crunching those numbers. Good luck, and may the momentum be with you!

To solve this problem, we can use the conservation of linear momentum, which states that the total linear momentum before an interaction is equal to the total linear momentum after the interaction.

Let's denote the momentum of the incident photon as p1, the momentum of the scattered photon as p2, and the momentum gained by the electron as Δp (change in momentum). Since the electron is initially at rest, its initial momentum is zero.

According to conservation of linear momentum, the total momentum before the interaction is equal to the total momentum after the interaction:

Total initial momentum = Total final momentum

p1 (momentum of incident photon) + 0 (initial momentum of electron) = p2 (momentum of scattered photon) + Δp (momentum gained by electron)

Since momentum (p) is given by the equation p = h/λ, where h is Planck's constant and λ is wavelength, we can rewrite the equation:

(h/λ1) + 0 = (h/λ2) + Δp

Substituting the given values:
λ1 = 0.2800 nm and λ2 = 0.2849 nm into the equation, and using the value of Planck's constant (h = 6.626 x 10^-34 J·s), we can solve for Δp.

Δp = (h/λ1) - (h/λ2)

Δp = (6.626 x 10^-34 J·s) / (0.2800 x 10^-9 m) - (6.626 x 10^-34 J·s) / (0.2849 x 10^-9 m)

Make sure to convert nanometers (nm) to meters (m) in order to use consistent units.

Calculating Δp, we get:
Δp ≈ 2.942 x 10^-27 kg·m/s

Therefore, the momentum gained by the electron is approximately 2.942 x 10^-27 kg·m/s.

Note: In the calculation, it's important to use consistent units (meters for length and kilograms for mass) in order to obtain the correct result.