Catching a Ruler

solved UniformAcceleration

Q. A friend lets go of $v_{0} = 0$ m/s a ruler held between your fingers, and you catch it after it has fallen $a = - g$ 21.0 cm $x_{1} = -21.0$ cm. Taking g as 9.8 m/s² $g = 9.8$ m/s², what is your reaction time, in seconds?

Inputs

gravity

g

9.8 m/s²

Outputs

reaction time

Δ t

How it's solved

Solved with UniformAcceleration

Given

$g = 9.8$ m/s²
$v_{0} = 0$ m/s from “lets go of”
$x_{1} = -21.0$ cm from “fallen” and “21.0 cm”
$a = - g$ from “fallen”

Assumed

$x_{0} = 0$ m

Find $Δ t$ .

Working

$x_{1} = -21.0$ cm $= −0.21$ m
fallen $a = - g$
$a = - 9.8$
$a = −9.8$ m/s²
position after a time(falling) $x_{1} = x_{0} + v_{0} \cdot Δ t + \frac{a \cdot {Δ t}^{2}}{2}$
$(−0.21) = 0 + 0 \cdot Δ t + \frac{(−9.8) \cdot {Δ t}^{2}}{2}$
$Δ t = 0.21 s$
$Δ t \approx 0.21$ sto 2 significant figures

Conclusion

The ruler falls freely from rest the whole time you take to react, so the distance it falls is a clock.

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 :catching_a_ruler do4  kind :numerical5  describes "A friend lets go of a ruler held between your fingers, and you catch it after it has " \6          "fallen 21.0 cm. Taking g as %{gravity}, %{reaction_time}, in seconds?"7  given gravity: "9.8 m/s2"8  asks reaction_time: "what is your reaction time"9end

solution.rb

1require_relative "../questions/catching_a_ruler"2require_relative "../science/motion_in_a_straight_line/uniform_acceleration"34solve :catching_a_ruler, using: UniformAcceleration do5  objects :ruler6  ruler.goes(:falling)78  take gravity: g9  assume ruler.initial_position => "0 m"10  deduce_from "lets go of", ruler.initial_velocity => "0 m/s"11  deduce_from "fallen", ruler.acceleration => -g12  deduce_from "fallen", "21.0 cm", ruler.final_position => "-21.0 cm"13  take reaction_time: falling.time_taken14  answer find_value(:reaction_time, in: "s", figures: 2)1516  conclusion "The ruler falls freely from rest the whole time you take to react, so the distance " \17             "it falls is a clock."18end

playground.rb

1require_relative "../solutions/catching_a_ruler"23play_with :catching_a_ruler do4  title "Catching a Ruler"5end

The laws it drew on

quantity	written	units	within	from
position	$x$	m	—	Velocity
displacement	$Δ x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average velocity	$v_{av}$	m/s	—	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
velocity	$v$	m/s	—	Acceleration
change of velocity	$Δ v$	m/s	—	Acceleration
average acceleration	$a_{av}$	m/s²	—	Acceleration
starting velocity	$v_{0}$	m/s	—	Acceleration
final velocity	$v$	m/s	—	Acceleration
acceleration	$a$	m/s²	—	UniformAcceleration
gravity	$g$	m/s²	—	UniformAcceleration
time	$t$	s	0 … 1e+05	UniformAcceleration
displacement of ruler during falling	$Δ x$	m	—
position of ruler at end of falling	$x_{1}$	m	—
position of ruler at start of falling	$x_{0}$	m	—
average velocity of ruler during falling	$v_{av}$	m/s	—
time taken of falling	$Δ t$	s	0 … 1e+05
average speed of ruler during falling	$c_{av}$	m/s	—
path length of ruler during falling	$d$	m	—
change of velocity of ruler during falling	$Δ v$	m/s	—
velocity of ruler at end of falling from ground	$v_{1}$	m/s	—
velocity of ruler at start of falling from ground	$v_{0}$	m/s	—
average acceleration of ruler during falling	$a_{av}$	m/s²	—
acceleration of ruler during falling	$a$	m/s²	—

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

v_{0} \equiv v_{i} (start of motion)

starting velocity

v \equiv v_{i} (end of motion)

final velocity

t \equiv Δ t_{m}

time

d_{i} (whole) = d_{i} (part 1) + d_{i} (part 2)

path in parts

Δ t_{w} = Δ t_{p} + Δ t_{p}

time in parts

a_{i} (part 1) = a_{i} (whole)

steady through its parts

Δ x = x_{1} - x_{0}

change of position(falling)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(falling)

c_{av} = \frac{d}{Δ t}

path a time(falling)

Δ v = v_{1} - v_{0}

velocity gained(falling)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(falling)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(falling)

x_{1} = x_{0} + v_{0} \cdot Δ t + \frac{a \cdot {Δ t}^{2}}{2}

position after a time(falling)

{v_{1}}^{2} = {v_{0}}^{2} + 2 \cdot a \cdot (x_{1} - x_{0})

velocity after a distance(falling)

x_{1} - x_{0} = \frac{(v_{1} + v_{0}) \cdot Δ t}{2}

mean of the velocities(falling)

a = - g

fallen