The Missing c — phryzby

solved DimensionalAnalysis

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?

How it's solved

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 web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

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

solution.rb

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

playground.rb

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

The laws it drew on

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