A Steady Push Forward

solved NewtonsSecondLaw

Q. A steady force on a body of 3.0 kg $m = 3.0$ kg takes it from 2.0 m/s $v_{0} = 2.0$ m/s to 3.5 m/s $v_{1} = 3.5$ m/s in 25 s $Δ t = 25$ s, without turning it. How strong is the force, in newtons?

Inputs

mass

m

3 kg

initial speed

v_{0}

2.0 m/s

final speed

v_{1}

3.5 m/s

time

Δ t

25.00 s

Outputs

force

F_{net}

How it's solved

Solved with NewtonsSecondLaw

Given

$m = 3.0$ kg
$v_{0} = 2.0$ m/s
$v_{1} = 3.5$ m/s
$Δ t = 25$ s

Find $F_{net}$ .

Working

velocity after a time(speeding up) $v_{1} = v_{0} + a \cdot Δ t$
$3.5 = 2 + a \cdot 25$
$a = 0.06$ m/s²
second law(speeding up) $F_{net} = m \cdot a$
$F_{net} = 3 \cdot 0.06$
$F_{net} = 0.18$ N
$F_{net} \approx 0.18$ Nto 2 significant figures

Conclusion

The force is the mass times the speed it gains each second, and it points the way the body is already going.

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 :a_steady_push_forward do4  kind :numerical5  describes "A steady force on a body of %{mass} takes it from %{initial_speed} to %{final_speed} in %{time}, without " \6            "turning it. %{force}, in newtons?"7  given mass: "3.0 kg"8  given initial_speed: "2.0 m/s"9  given final_speed: "3.5 m/s"10  given time: "25 s"11  asks force: "How strong is the force"12end

solution.rb

1require_relative "../questions/a_steady_push_forward"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :a_steady_push_forward, using: NewtonsSecondLaw do5  objects :body6  body.goes(:speeding_up)78  take mass: body.mass9  take initial_speed: body.initial_velocity10  take final_speed: body.final_velocity11  take time: speeding_up.time_taken1213  force = body.net_force14  answer force:, in: "N", figures: 21516  conclusion "The force is the mass times the speed it gains each second, and it points the way " \17             "the body is already going."18end

playground.rb

1require_relative "../solutions/a_steady_push_forward"23play_with :a_steady_push_forward do4  title "A Steady Push Forward"5end

The laws it drew on

quantity	written	units	within	from
volume	$volume$	—	—	Vectors
mass	$m$	kg	0 … ∞	Vectors
speed	$speed$	—	—	Vectors
density	$density$	—	—	Vectors
number of moles	$n$	—	—	Vectors
angular frequency	${angular}_{frequency}$	—	—	Vectors
work	$work$	—	—	Vectors
current	$current$	—	—	Vectors
temperature	$temperature$	—	—	Vectors
pressure	$pressure$	—	—	Vectors
time	$t$	s	0 … 1e+05	Vectors
power	$power$	—	—	Vectors
total path length	$s$	—	—	Vectors
energy	$energy$	—	—	Vectors
gravitational potential	${gravitational}_{potential}$	—	—	Vectors
coefficient of friction	$μ$	—	—	Vectors
charge	$charge$	—	—	Vectors
acceleration	$a$	m/s²	—	Vectors
velocity	$v$	m/s	—	Vectors
displacement	$Δ x$	m	—	Vectors
angular velocity	${angular}_{velocity}$	—	—	Vectors
force	$F$	N	—	Vectors
angular momentum	${angular}_{momentum}$	—	—	Vectors
linear momentum	$p_{lin}$	—	—	Vectors
electric field	${electric}_{field}$	—	—	Vectors
average velocity	$v_{av}$	m/s	—	Vectors
magnetic moment	${magnetic}_{moment}$	—	—	Vectors
relative velocity	$v_{AB}$	—	—	Vectors
impulse	$impulse$	—	—	Vectors
first vector	$A$	—	0 … ∞	Vectors
second vector	$B$	—	0 … ∞	Vectors
angle between	$φ$	—	0 … 3.142	Vectors
resultant	$R$	—	0 … ∞	Vectors
resultant direction	$β$	—	0 … 3.142	Vectors
direction	$α$	—	-3.142 … 3.142	Vectors
x component	$A_{x}$	—	—	Vectors
y component	$A_{y}$	—	—	Vectors
position	$x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
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
gravity	$g$	m/s²	—	UniformAcceleration
net force	$F_{net}$	N	—	NewtonsSecondLaw
weight	$W$	N	—	NewtonsSecondLaw
displacement of body during speeding up	$Δ x$	m	—
position of body at end of speeding up	$x_{1}$	m	—
position of body at start of speeding up	$x_{0}$	m	—
average velocity of body during speeding up	$v_{av}$	m/s	—
time taken of speeding up	$Δ t$	s	0 … 1e+05
average speed of body during speeding up	$c_{av}$	m/s	—
path length of body during speeding up	$d$	m	—
change of velocity of body during speeding up	$Δ v$	m/s	—
velocity of body at end of speeding up from ground	$v_{1}$	m/s	—
velocity of body at start of speeding up from ground	$v_{0}$	m/s	—
average acceleration of body during speeding up	$a_{av}$	m/s²	—
acceleration of body during speeding up	$a$	m/s²	—
net force on body during speeding up	$F_{net}$	N	—
mass of body	$m$	kg	0 … ∞
weight on ground	$W_{g}$	N	—
weight on body	$W$	N	—

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

A = \sqrt{{A_{x}}^{2} + {A_{y}}^{2}}

magnitude from components

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

m_{b} \neq 0

has mass?

R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ

law of cosines

R \cdot \sin β = B \cdot \sin φ

law of sines

R \cdot \cos β = A + B \cdot \cos φ

along the first

A_{x} = A \cdot \cos α

resolved along x

A_{y} = A \cdot \sin α

resolved along y

Δ x = x_{1} - x_{0}

change of position(speeding up)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(speeding up)

c_{av} = \frac{d}{Δ t}

path a time(speeding up)

Δ v = v_{1} - v_{0}

velocity gained(speeding up)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(speeding up)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(speeding up)

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

position after a time(speeding up)

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

velocity after a distance(speeding up)

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

mean of the velocities(speeding up)

F_{net} = m \cdot a

second law(speeding up)

m \neq 0

has mass?(body)

F_{net} = 0

free of forces?(body, speeding up)