I don't get this problem! Please show work. Thanks!!
The de Broglie wavelength of a proton in a particle accelerator is 3.60 x 10^-14 m. Determine the kinetic energy (in joules) of the proton.
To determine the kinetic energy of the proton, we need to use the de Broglie wavelength formula.
The de Broglie wavelength (λ) of a particle is related to its momentum (p) through the equation:
λ = h / p
where λ is the de Broglie wavelength, h is the Planck's constant (6.63 x 10^-34 J·s), and p is the momentum of the particle.
In this problem, we are given the de Broglie wavelength (λ = 3.60 x 10^-14 m). We can rearrange the equation to solve for momentum:
λ = h / p
p = h / λ
Substituting the given values:
p = (6.63 x 10^-34 J·s) / (3.60 x 10^-14 m)
Now we can calculate the momentum of the proton.
p = (6.63 x 10^-34 J·s) / (3.60 x 10^-14 m)
p ≈ 1.84 x 10^-20 kg·m/s
The kinetic energy (KE) of a particle is related to its momentum through the equation:
KE = p^2 / (2m)
where KE is the kinetic energy, p is the momentum of the particle, and m is the mass of the particle.
The mass of a proton (m) is approximately 1.67 x 10^-27 kg.
Substituting the values:
KE = (1.84 x 10^-20 kg·m/s)^2 / (2 * 1.67 x 10^-27 kg)
Now we can calculate the kinetic energy of the proton.
KE = (1.84 x 10^-20 kg·m/s)^2 / (2 * 1.67 x 10^-27 kg)
KE ≈ 2.12 x 10^-10 J
Therefore, the kinetic energy of the proton is approximately 2.12 x 10^-10 Joules.
To determine the kinetic energy of the proton, we can make use of the de Broglie wavelength equation:
λ = h / p
Where λ represents the de Broglie wavelength, h is the Planck's constant (h = 6.626 x 10^(-34) J·s), and p is the momentum of the proton.
The momentum of a particle is given by:
p = m * v
Where p is the momentum, m is the mass of the proton, and v is the velocity of the proton.
First, we need to find the velocity of the proton. We can find it using the de Broglie wavelength equation. Rearranging the equation, we have:
v = λ * h / m
Substituting the given de Broglie wavelength (3.60 x 10^(-14) m) and the mass of a proton (m = 1.67 x 10^(-27) kg), we get:
v = (3.60 x 10^(-14) m) * (6.626 x 10^(-34) J·s) / (1.67 x 10^(-27) kg)
Now, we can calculate the velocity of the proton.
Finally, we can determine the kinetic energy of the proton using the formula:
Kinetic Energy = 0.5 * m * v^2
Substituting the calculated velocity of the proton and the mass of the proton into the equation, we can solve for the kinetic energy in joules.