A certain sports utility vehicle is traveling at a speed of $57 mph$ $"57 mph"$ . If its mass is $5400 lbs$ $"5400 lbs"$ , what is its de Broglie wavelength?

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A certain sports utility vehicle is traveling at a speed of $57 mph$ $"57 mph"$ . If its mass is $5400 lbs$ $"5400 lbs"$ , what is its de Broglie wavelength?

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Answer 1 · 2017-09-07T00:28:03

$λ = 1.1 \times 10^{- 38} m$ $lambda = 1.1 xx 10^(-38) "m"$

The de Broglie wavelength $λ$ $lambda$ in $m$ $"m"$ is given by the de Broglie relation:

$λ = \frac{h}{m v}$ $lambda = h/(mv)$

where:

$h = 6.626 \times 10^{- 34} J \cdot s$ $h = 6.626 xx 10^(-34) "J" cdot"s"$ is Planck's constant.

$m$ $m$ is the mass of the mass-ive object in $kg$ $"kg"$ .

$v$ $v$ is its velocity in $m/s$ $"m/s"$ .

The units here are wacky, so we'll need to convert the $lb$ $"lb"$ into $kg$ $"kg"$ and the $mph$ $"mph"$ into $m/s$ $"m/s"$ . We have the following conversion factors to consider:

$\frac{2.2 lb}{kg} \frac{5280 ft}{1 mi} \frac{12 in}{1 ft}$ $"2.2 lb"/"kg"" "" "" ""5280 ft"/"1 mi"" "" ""12 in"/"1 ft"$

$\frac{2.54 cm}{1 in} \frac{100 cm}{1 m} \frac{60 min}{1 hr}$ $"2.54 cm"/"1 in"" "" ""100 cm"/"1 m"" "" ""60 min"/"1 hr"$

$\frac{60 s}{1 min}$ $"60 s"/"1 min"$

First, convert the mass:

$5400 cancel"lbs" xx "1 kg"/(2.2 cancel"lbs") = "2454.5 kg"$
$" "" "" "" "" "" "" "" "$ (we'll round later)

Now convert the velocity:

$(57 cancel"mi")/cancel"hr" xx (5280 cancel"ft")/(cancel"1 mi") xx (12 cancel"in")/(cancel"1 ft") xx (2.54 cancel"cm")/(cancel"1 in") xx "1 m"/(100 cancel"cm") xx (cancel"1 hr")/(60 cancel"min") xx (cancel"1 min")/("60 s")$

$=$ $"25.481 m/s"$
(we'll round later)

Lastly, just solve the de Broglie relation for the wavelength.

$color(blue)(lambda) = (6.626 xx 10^(-34) cancel"kg" cdot"m"^cancel(2)"/"cancel"s")/(2454.5 cancel"kg" cdot 25.481 cancel("m/s"))$

$= ulcolor(blue)(1.1 xx 10^(-38) "m")$

And this number is justifiably puny.

A macroscopic particle like a sports utility vehicle has practically no wave characteristics to speak of. Only really light particles moving at very fast speeds are quantum mechanical.

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A certain sports utility vehicle is traveling at a speed of $57 mph$ $"57 mph"$ . If its mass is $5400 lbs$ $"5400 lbs"$ , what is its de Broglie wavelength?

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A certain sports utility vehicle is traveling at a speed of $57 mph$ $"57 mph"$ . If its mass is $5400 lbs$ $"5400 lbs"$ , what is its de Broglie wavelength?