Calculate the wavelength of
(a) a 140 g ball moving at 40 m/s
(b) a proton moving at the same speed
(c) an electron moving at the same speed.

(Answer: (a) 1.18 ï½˜ 10⁻³⁴ m  (b) 9.92 nm (c) 1.82 ｘ 10⁻⁵ m)



Given Data:

The velocity of the ball = Velocity of the proton = Velocity of the electron = v = 40 m s⁻¹

Mass of the ball = `\m_b` = 140 g = 0.14 kg

Mass of proton = `\m_p` = 1.67 ｘ 10⁻²⁷ kg

Mass of electron = `\m_e` = 9.11 ｘ 10⁻³¹ kg


∴ Plank's Constant = h =  6.626 ｘ 10⁻³⁴ m² kg s⁻¹

To Find:

(a) Wavelength of ball = `\λ_b` =  ? 

(b) Wavelength of ball = `\λ_p` =  ? 

(c) Wavelength of ball = `\λ_n` =  ? 



Solution:


We know that momentum P is given by

P = `\frac {h}{λ}`


or


λ = `\frac {h}{P}`


but P = mv , So


λ = `\frac {h}{mv}` ------------(1)


(a) Wavelength of ball = `\λ_b` =  ? 

Using equation (1)

`\λ_b` = `\frac {h}{m_b v}`

by putting values

`\λ_b` = `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹}{0.14 kg ｘ 40 m s⁻¹ }`

`\λ_b` = `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹}{5.6 kg m s⁻¹ }`

`\λ_b` = 1.193 ï½˜ 10⁻³⁴ m --------------Ans. 1



(b) Wavelength of Proton = `\λ_p` =  ? 

Using equation (1)

`\λ_p` = `\frac {h}{m_p v}`

by putting values

`\λ_p` = `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹}{1.67 ｘ 10⁻²⁷ kg ｘ 40 m s⁻¹ }`

`\λ_p` = `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹}{66.8 ｘ 10⁻²⁷ kg m s⁻¹ }`

`\λ_p` = 0.0992 ï½˜ 10⁻⁷ m

or

`\λ_p` = 9.92 ï½˜ 10⁻⁹ m 

or

`\λ_p` = 9.92 nm --------------Ans. 2


(c) Wavelength of Electron = `\λ_e` =  ? 

Using equation (1)

`\λ_e` = `\frac {h}{m_e v}`

by putting values

`\λ_e` = `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹}{9.11 ｘ 10⁻³¹ kg ｘ 40 m s⁻¹ }`

`\λ_e` = `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹}{364.4 ｘ 10⁻³¹ kg m s⁻¹ }`

`\λ_e` = 0.01818 ï½˜ 10⁻³ m

or

`\λ_e` = 1.818 ï½˜ 10⁻⁵ m --------------Ans. 3




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