Along a Straight Line

solved UniformAcceleration

Q. A body starts at the origin $x_{0} = 0$ m with velocity $v_{0}$ and speeds up or slows down steadily at $a$ . Where is it after $t$ , and how fast is it going?

Inputs

starting velocity

v_{0}

10.0 m/s

acceleration

a

-2 m/s²

time

t

3.00 s

Outputs

position

x

21.00 m

velocity

v

4.0 m/s

How it's solved

Solved with UniformAcceleration

Given

$x_{0} = 0$ m from “the origin”
$v_{0}$ , $a$ , $t$ yours to vary

Find $x$ , $v$ .

Working

position after a time $x = x_{0} + v_{0} \cdot t + \frac{a \cdot t^{2}}{2}$
$x = 0 + 10 \cdot 3 + \frac{(−2) \cdot 3^{2}}{2}$
$x = 21$ m
velocity after a time $v = v_{0} + a \cdot t$
$v = 10 + (−2) \cdot 3$
$v = 4$ m/s

Conclusion

Where the steady change of velocity has taken it: its velocity changes by the same amount every second, and its position by that velocity, second after second.

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 :along_a_straight_line do4  describes "A body starts at the origin with velocity %{starting_velocity} and speeds up or slows down steadily at %{acceleration}. " \5          "%{position} after %{time}, and %{velocity}?"6  given :starting_velocity7  given :acceleration8  given :time9  asks position: "Where is it", velocity: "how fast is it going"10end

solution.rb

1require_relative "../questions/along_a_straight_line"2require_relative "../science/motion_in_a_straight_line/uniform_acceleration"34solve :along_a_straight_line, using: UniformAcceleration do5  take starting_velocity: :v_0, acceleration: :a, time: :t, position: :x, velocity: :v6  deduce_from "the origin", x_0: "0 m"7  find_value :position8  find_value :velocity910  conclusion "Where the steady change of velocity has taken it: its velocity changes by the same " \11             "amount every second, and its position by that velocity, second after second."12end

playground.rb

1require_relative "../solutions/along_a_straight_line"23play_with :along_a_straight_line do4  title "Along a Straight Line"56  input :starting_velocity, -20.0..20.0, in: "m/s", step: 0.5, default: 10.0, as: "starting velocity"7  input :acceleration,      -10.0..10.0, in: "m/s2", step: 0.5, default: -2.0, as: "acceleration"8  input :time,              0.0..10.0,   in: "s",    step: 0.1, default: 3.0,  as: "time"910  output :position, in: "m"11  output :velocity, in: "m/s"12end

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

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 - x_{0}

change of position

v_{av} = \frac{Δ x}{Δ t}

displacement a time

c_{av} = \frac{d}{Δ t}

path a time

Δ v = v - v_{0}

velocity gained

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time

v = v_{0} + a \cdot t

velocity after a time

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

position after a time

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

velocity after a distance

x - x_{0} = \frac{(v + v_{0}) \cdot t}{2}

mean of the velocities