Along a Straight Line

Q. A body starts at the origin with velocity v₀ and speeds up or slows down steadily at a. Where is it after t, and how fast is it going?

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

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?

The playground

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

The solution

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 Ruby

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

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

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

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