Ruled Out by Dimensions

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. expects B

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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.

The playground

The solution

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 Ruby

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

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

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