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QuestionFind the de Broglie wavelength of a 142 g142 \mathrm{~g} ball moving at 42.36 m/s42.36 \mathrm{~m/s}. Round to 3 sig figs.

Studdy Solution

STEP 1

Assumptions1. The mass of the ball is 142 g142 \mathrm{~g} . The speed of the ball is 42.36 m/s42.36 \mathrm{~m} / \mathrm{s}
3. The ball can be modeled as a single particle4. We use the de Broglie equation to calculate the wavelength5. The Planck constant (hh) is 6.62607015×10−34 m kg/s6.62607015 \times10^{-34} \mathrm{~m}^ \mathrm{~kg} / \mathrm{s}

STEP 2

First, we need to convert the mass of the ball from grams to kilograms, since the Planck constant is given in terms of kilograms. We can do this by multiplying the mass in grams by the conversion factor 1 kg/1000 g1 \mathrm{~kg} /1000 \mathrm{~g}.
Masskg=Massg×1 kg1000 gMass_{kg} = Mass_{g} \times \frac{1 \mathrm{~kg}}{1000 \mathrm{~g}}

STEP 3

Now, plug in the given value for the mass in grams to calculate the mass in kilograms.
Masskg=142 g×1 kg1000 gMass_{kg} =142 \mathrm{~g} \times \frac{1 \mathrm{~kg}}{1000 \mathrm{~g}}

STEP 4

Calculate the mass in kilograms.
Masskg=142 g×1 kg1000 g=0.142 kgMass_{kg} =142 \mathrm{~g} \times \frac{1 \mathrm{~kg}}{1000 \mathrm{~g}} =0.142 \mathrm{~kg}

STEP 5

Now that we have the mass in kilograms, we can use the de Broglie equation to calculate the wavelength. The de Broglie equation is given byλ=hmv\lambda = \frac{h}{mv}where λ\lambda is the wavelength, hh is the Planck constant, mm is the mass, and vv is the speed.

STEP 6

Plug in the values for the Planck constant, the mass, and the speed into the de Broglie equation to calculate the wavelength.
λ=6.62607015×10−34 m2 kg/s0.142 kg×42.36 m/s\lambda = \frac{6.62607015 \times10^{-34} \mathrm{~m}^2 \mathrm{~kg} / \mathrm{s}}{0.142 \mathrm{~kg} \times42.36 \mathrm{~m} / \mathrm{s}}

STEP 7

Calculate the wavelength.
λ=6.62607015×10−34 m2 kg/s0.142 kg×42.36 m/s=1.10×10−34 m\lambda = \frac{6.62607015 \times10^{-34} \mathrm{~m}^2 \mathrm{~kg} / \mathrm{s}}{0.142 \mathrm{~kg} \times42.36 \mathrm{~m} / \mathrm{s}} =1.10 \times10^{-34} \mathrm{~m}The de Broglie wavelength of the ball is 1.10×10−34 m1.10 \times10^{-34} \mathrm{~m} to3 significant figures.

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