To determine the speed of a proton that has a de Broglie wavelength of 129 picometers (pm), we need to use the de Broglie equation which links the momentum of a particle with its wavelength. The de Broglie wavelength [tex]\( \lambda \)[/tex] for a particle with momentum [tex]\( p \)[/tex] is given by:
[tex]\[ \lambda = \frac{h}{p} \][/tex]
where:
- [tex]\( \lambda \)[/tex] is the de Broglie wavelength,
- [tex]\( h \)[/tex] is Planck's constant ([tex]\( 6.626 \times 10^{-34} \)[/tex] Js),
- [tex]\( p \)[/tex] is the momentum of the particle.
The momentum [tex]\( p \)[/tex] of a particle with mass [tex]\( m \)[/tex] and velocity [tex]\( v \)[/tex] is given by:
[tex]\[ p = mv \][/tex]
Therefore, substituting [tex]\( p = mv \)[/tex] into the de Broglie equation, we get:
[tex]\[ \lambda = \frac{h}{mv} \][/tex]
Solving for the velocity [tex]\( v \)[/tex], we have:
[tex]\[ v = \frac{h}{m \lambda} \][/tex]
Given:
- Planck's constant, [tex]\( h = 6.626 \times 10^{-34} \)[/tex] Js,
- Proton mass, [tex]\( m = 1.673 \times 10^{-27} \)[/tex] kg,
- de Broglie wavelength, [tex]\( \lambda = 129 \)[/tex] pm [tex]\( = 129 \times 10^{-12} \)[/tex] meters,
We can now plug these values into the equation to find the speed [tex]\( v \)[/tex]:
[tex]\[ v = \frac{6.626 \times 10^{-34}}{1.673 \times 10^{-27} \times 129 \times 10^{-12}} \][/tex]
When we substitute these values and perform the calculation:
[tex]\[ v \approx 3070.193728946283 \][/tex]
Thus, the speed of the proton is approximately [tex]\( 3070.19 \)[/tex] meters per second.
