Calculate the wavelengths of the following objects in nm

Question

1. an 80.0 kg athlete running a "4-minute mile" (i.e. 4.00 min/mile) 
2. Earth (mass = 6.20 × 10²⁷ g) moving through space at 2.80 × 10⁴ m/s

Answer 1

Hello Daisy! Thanks for your question.

This question requires you to use the de broglie equation which relates wavelength and momentum.

The de broglie equation says wavelength (λ) is equal to Planck's constant (h) divided by momentum (p).

λ = h/p de broglie equation

Planck's constant (h) is 6.626 x 10^-34 Js.

Momentum (p) is equal to velocity (v) times mass (m).

p = vm momentum

Combining both those equations, you get the following equation:

λ= h/(vm)

So, now just put in your variables. But you must be careful to make sure your units all match. A joule (J) is equal to a kg*m²/s². So, mass has to be in kg and velocity has to be in m/s in order to use Planck's constant in Js. That also means the wavelength (λ) will be in meters.

1. an 80.0 kg athlete running a "4-minute mile" (i.e. 4.00 min/mile) 

Convert 4.00 min/mile to m/s. Notice m/s is distance/time, so you have to start by flipping min/mile to mile/min.

1mile/4.00 min x 1min/60s x 1610 m/1mile = 6.71 m/s

v= 6.71 m/s

m= 80.0 kg

h= 6.626 x 10^-34 Js

λ= h/(vm)

λ= 6.626 x 10^-34 Js/(6.71 m/s * 80.0 kg)

λ= 6.626 x 10^-34 kg*m²/s /(6.71 m/s * 80.0 kg)

λ= 1.23 x 10^-36 m

Now, we need to convert m to nm. 1 nm is equal to 10^-9 m.

1.23x10^-36 m x 1nm/10^-9m = 1.23x10^-27nm

2.  Earth (mass = 6.20 × 10²⁷ g) moving through space at 2.80 × 10⁴ m/s.

Convert mass from g to kg. There are 1000 g in 1 kg.

6.20 × 10²⁷ g x 1 kg/1000 g = 6.20 × 10²⁴kg

v= 2.80 × 10⁴  m/s

m= 6.20 × 10²⁴kg

h= 6.626 x 10^-34 Js

λ= h/(vm)

λ= 6.626 x 10^-34 Js/(2.80 × 10⁴  m/s * 6.20 × 10²⁴kg)

λ= 6.626 x 10^-34 kg*m²/s /(2.80 × 10⁴  m/s * 6.20 × 10²⁴kg)

λ= 3.84 x 10^-63 m

Now, convert meters to nanometers.

3.84 x 10^-63 m x 1nm/10^-9m = 3.84 x 10^-54 nm