Ruled Out by Dimensions

solved DimensionalAnalysis

Q. A body of mass m moves at speed v with acceleration a. Which of these could be its kinetic energy K? Dimensions rule out all but one.

How it's solved

Solved with DimensionalAnalysis

Working

$K = m^{2} \cdot v^{3}$ : one side is [M L² T⁻²] and the other [M² L³ T⁻³]
$K = \frac{m \cdot v^{2}}{2}$ : both sides are [M L² T⁻²]
$K = m \cdot a$ : one side is [M L² T⁻²] and the other [M L T⁻²]
$K = \frac{m \cdot v^{2}}{2} + m \cdot a$ : cannot add [M L T⁻²] to [M L² T⁻²]

Conclusion

A formula without the dimensions of energy cannot be the kinetic energy, and a sum of unlike quantities is no quantity at all. Dimensions alone could not tell the ½ from any other number.

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 :ruled_out_by_dimensions do4  kind :scq5  describes "A body of mass m moves at speed v with acceleration a. Which of these could be its " \6          "kinetic energy K? Dimensions rule out all but one."78  option "K = m²v³", as: :mass_squared9  option "K = ½mv²", as: :half_mv_squared10  option "K = ma", as: :ma11  option "K = ½mv² + ma", as: :sum12end

solution.rb

1require_relative "../questions/ruled_out_by_dimensions"2require_relative "../science/units_and_measurement/dimensional_analysis"34solve :ruled_out_by_dimensions, using: DimensionalAnalysis do5  take mass_squared: K == m**2 * v**3,6       half_mv_squared: K == m * v**2 / 2,7       ma: K == m * a,8       sum: K == m * v**2 / 2 + m * a9  answer options.select { |o| consistent?(o.taken) }1011  conclusion "A formula without the dimensions of energy cannot be the kinetic energy, and a sum " \12             "of unlike quantities is no quantity at all. Dimensions alone could not tell the ½ " \13             "from any other number."14end

playground.rb

1require_relative "../solutions/ruled_out_by_dimensions"23play_with :ruled_out_by_dimensions do4  title "Ruled Out by Dimensions"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