Respuesta :
Answer:
Probability of tunneling is [tex]10^{- 1.17\times 10^{32}}[/tex]
Solution:
As per the question:
Velocity of the tennis ball, v = 120 mph = 54 m/s
Mass of the tennis ball, m = 100 g = 0.1 kg
Thickness of the tennis ball, t = 2.0 mm = [tex]2.0\times 10^{- 3}\ m[/tex]
Max velocity of the tennis ball, [tex]v_{m} = 200\ mph[/tex] = 89 m/s
Now,
The maximum kinetic energy of the tennis ball is given by:
[tex]KE = \frac{1}{2}mv_{m}^{2} = \frac{1}{2}\times 0.1\times 89^{2} = 396.05\ J[/tex]
Kinetic energy of the tennis ball, KE' = [tex]\frac{1}{2}mv^{2} = 0.5\times 0.1\times 54^{2} = 154.8\ m/s[/tex]
Now, the distance the ball can penetrate to is given by:
[tex]\eta = \frac{\bar{h}}{\sqrt{2m(KE - KE')}}[/tex]
[tex]\bar{h} = \frac{h}{2\pi} = \frac{6.626\times 10^{- 34}}{2\pi} = 1.0545\times 10^{- 34}\ Js[/tex]
Thus
[tex]\eta = \frac{1.0545\times 10^{- 34}}{\sqrt{2\times 0.1(396.05 - 154.8)}}[/tex]
[tex]\eta = \frac{1.0545\times 10^{- 34}}{\sqrt{2\times 0.1(396.05 - 154.8)}}[/tex]
[tex]\eta = 1.52\times 10^{-35}\ m[/tex]
Now,
We can calculate the tunneling probability as:
[tex]P(t) = e^{\frac{- 2t}{\eta}}[/tex]
[tex]P(t) = e^{\frac{- 2\times 2.0\times 10^{- 3}}{1.52\times 10^{-35}}} = e^{-2.63\times 10^{32}}[/tex]
[tex]P(t) = e^{-2.63\times 10^{32}}[/tex]
Taking log on both the sides:
[tex]logP(t) = -2.63\times 10^{32} loge[/tex]
[tex]P(t) = 10^{- 1.17\times 10^{32}}[/tex]
