A 227−lb fullback runs the 40−yd dash at a speed of 17.25 pm 0.10 mi/h. Enter your answers in scientific notation. (a) What is his de Broglie wavelength (in meters)? × 10 (select) m (b) What is the uncertainty in his position based on the de Broglie wavelength?

His de Broglie wavelength is 8.35×10⁻³⁷ m

The uncertainty in his position is 1.15 × 10⁻³⁵ m

Explanation:

First, Convert 227-lb to kg and convert the unit of the speed from mi/h to m/s.

To convert 227-lb to kg,

1-lb = 0.453592 kg

∴ 227-lb = 227 ×  0.453592 kg

227-lb = 102.97 kg

To convert 17.25 ± 0.10 mi/h to m/s

1 mi = 1609.34 m

and 1 h = 3600 s

Therefore,

17.25 mi/h = (17.25 ×1609.34)/3600 m/s = 7.71 m/s

and 0.10 mi/h =  (0.10 ×1609.34)/3600 m/s = 0.044704 m/s

Hence, the speed 17.25 ± 0.10 mi/h = 7.71 ± 0.044704 m/s

Now

(a) To determine the de Broglie wavelength,

De Broglie wavelength is given by

λ = h/mv

Where λ is the de Broglie wavelength

h is Planck's constant (h = 6.626×10⁻³⁴ kgm²/s)

m is the mass

and v is the speed (velocity)

From the question

m = 102.97 kg

v = 7.71 m/s

Therefore,

λ = 6.626×10⁻³⁴ / (102.97×7.71)

λ = 8.35×10⁻³⁷ m

Hence, his de Broglie wavelength is 8.35×10⁻³⁷ m

(b) To calculate the uncertainty in his position

From

Δx = h/(4πmΔv)

Where Δx is the uncertainty in the position

h is Planck's constant (h = 6.626×10⁻³⁴ kgm²/s)

π is a constant ( π = 3.14)

m is the mass

Δv is the uncertainty in speed

From the question

m = 102.97 kg

Δv = 0.044704 m/s

Hence,

Δx = 6.626×10⁻³⁴  / (4×3.14×102.97×0.044704)

Δx = 1.15 × 10⁻³⁵ m

Hence, the uncertainty in his position is 1.15 × 10⁻³⁵ m.

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