The Missing c — phryzby

Q. In units where light's speed c is 1, a moving body's mass is written m = m₀ ⁄ √(1 − v²). Put c back. Which is right? expects A

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Q. In units where light's speed c is 1, a moving body's mass is written m = m₀ ⁄ √(1 − v²). Put c back. Which is right?

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The solution

Solved with DimensionalAnalysis

Working

$m = \frac{m_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$ : both sides are [M]
$m = \frac{m_{0}}{\sqrt{1 - v^{2} \cdot c^{2}}}$ : cannot take [L⁴ T⁻⁴] from [M⁰ L⁰ T⁰]
$m = \frac{m_{0}}{\sqrt{c^{2} - v^{2}}}$ : one side is [M] and the other [M L⁻¹ T]
$m = \frac{m_{0} \cdot c}{\sqrt{1 - v^{2}}}$ : cannot take [L² T⁻²] from [M⁰ L⁰ T⁰]

Conclusion

Only a pure number can be taken from 1, so v² must be divided by something of the same dimensions: c². And m and m₀ are both masses, so no c can be left over outside the root.

The Ruby

1require_relative "../physics"23question :the_missing_c do4  kind :scq5  describes "In units where light's speed c is 1, a moving body's mass is written m = m₀ ⁄ √(1 − v²). " \6          "Put c back. Which is right?"78  option "m = m₀ ⁄ √(1 − v²⁄c²)", as: :over_c_squared9  option "m = m₀ ⁄ √(1 − v²c²)", as: :times_c_squared10  option "m = m₀ ⁄ √(c² − v²)", as: :c_squared_less11  option "m = m₀ c ⁄ √(1 − v²)", as: :c_outside12end

1require_relative "../questions/the_missing_c"2require_relative "../science/units_and_measurement/dimensional_analysis"34solve :the_missing_c, using: DimensionalAnalysis do5  take over_c_squared: m == m_0 / sqrt(1 - v**2 / c**2),6       times_c_squared: m == m_0 / sqrt(1 - v**2 * c**2),7       c_squared_less: m == m_0 / sqrt(c**2 - v**2),8       c_outside: m == m_0 * c / sqrt(1 - v**2)9  answer options.select { |o| consistent?(o.taken) }1011  conclusion "Only a pure number can be taken from 1, so v² must be divided by something of the " \12             "same dimensions: c². And m and m₀ are both masses, so no c can be left over outside " \13             "the root."14end

1require_relative "../solutions/the_missing_c"23play_with :the_missing_c do4  title "The Missing c"5end

The laws

quantity	written	units	within	from
pi	$π$	—	0 … ∞	DimensionalAnalysis
speed of light	$c$	m/s	—	DimensionalAnalysis
length	$l$	m	—	DimensionalAnalysis
gravity	$g$	m/s²	—	DimensionalAnalysis
mass	$m$	kg	—	DimensionalAnalysis
period	$T$	s	—	DimensionalAnalysis
constant	$k$	—	—	DimensionalAnalysis
power of length	$x$	—	—	DimensionalAnalysis
power of gravity	$y$	—	—	DimensionalAnalysis
power of mass	$z$	—	—	DimensionalAnalysis
speed	$v$	m/s	0 … ∞	DimensionalAnalysis
height	$h$	m	—	DimensionalAnalysis
time	$t$	s	0 … ∞	DimensionalAnalysis
acceleration	$a$	m/s²	—	DimensionalAnalysis
start	$s_{0}$	m	—	DimensionalAnalysis
starting speed	$v_{0}$	m/s	—	DimensionalAnalysis
distance	$s$	m	—	DimensionalAnalysis
rest mass	$m_{0}$	kg	—	DimensionalAnalysis
kinetic energy	$K$	J	—	DimensionalAnalysis

T = k \cdot l^{x} \cdot g^{y} \cdot m^{z}

period as powers

x + y = 0

matching length

-2 \cdot y = 1

matching time

z = 0

matching mass

\frac{m \cdot v^{2}}{2} = m \cdot g \cdot h

energy conserved

s = s_{0} + v_{0} \cdot t + \frac{a \cdot t^{2}}{2}

distance under constant acceleration