Stopped by a Steady Force

Q. A body of 20 kg is moving at 15 m/s when a steady force of 50 N starts to act against its motion. How long does it take to stop, in seconds? expects 6.00 s

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The web of concepts

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The equations it used

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Q. A body of 20 kg $m = 20$ kg is moving at 15 m/s $v_{0} = 15$ m/s when a steady force of 50 N $| F_{net} | = 50$ N starts to act against its motion $F_{net} = - | F_{net} |$ . How long does it take to stop $v_{1} = 0$ m/s, in seconds?

The playground

Inputs

mass

m

20 kg

speed

v_{0}

15.0 m/s

force

| F_{net} |

50 N

Outputs

time

Δ t

The solution

Solved with NewtonsSecondLaw

Given

$m = 20$ kg
$v_{0} = 15$ m/s
$| F_{net} | = 50$ N
$v_{1} = 0$ m/s from “to stop”
$F_{net} = - | F_{net} |$ from “against its motion”

Find $Δ t$ .

Working

against its motion $F_{net} = - | F_{net} |$
$F_{net} = - 50$
$F_{net} = −50$ N
second law(stopping, x) $F_{net} = m \cdot a$
$(−50) = 20 \cdot a$
$a = −2.5$ m/s²
velocity after a time(stopping, x) $v_{1} = v_{0} + a \cdot Δ t$
$0 = 15 + (−2.5) \cdot Δ t$
$Δ t = 6.00 s$

Conclusion

The force slows it at a steady rate, the force over the mass, so the time to stop is the speed it has to lose over that rate.

The Ruby

1require_relative "../physics"23question :stopped_by_a_steady_force do4  kind :numerical5  describes "A body of %{mass} is moving at %{speed} when a steady force of %{force} starts to act against " \6            "its motion. %{time}, in seconds?"7  given mass: "20 kg"8  given speed: "15 m/s"9  given force: "50 N"10  asks time: "How long does it take to stop"11end

1require_relative "../questions/stopped_by_a_steady_force"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :stopped_by_a_steady_force, using: NewtonsSecondLaw do5  objects :body6  axes :x7  body.goes(:stopping)89  take mass: body.mass10  take speed: body.initial_velocity11  take force: body.net_force.magnitude1213  deduce_from "against its motion", body.net_force.direction => -x14  deduce_from "to stop", body.final_velocity => "0 m/s"1516  time = stopping.time_taken17  answer time:, in: "s"1819  conclusion "The force slows it at a steady rate, the force over the mass, so the time to stop is " \20             "the speed it has to lose over that rate."21end

1require_relative "../solutions/stopped_by_a_steady_force"23play_with :stopped_by_a_steady_force do4  title "Stopped by a Steady Force"5end

The laws

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 stopping along x	$Δ x$	m	—
position of body at end of stopping along x	$x_{1}$	m	—
position of body at start of stopping along x	$x_{0}$	m	—
average velocity of body during stopping along x	$v_{av}$	m/s	—
time taken of stopping	$Δ t$	s	0 … 1e+05
average speed of body during stopping	$c_{av}$	m/s	—
path length of body during stopping	$d$	m	—
change of velocity of body during stopping along x	$Δ v$	m/s	—
velocity of body at end of stopping from ground along x	$v_{1}$	m/s	—
velocity of body at start of stopping from ground along x	$v_{0}$	m/s	—
average acceleration of body during stopping along x	$a_{av}$	m/s²	—
acceleration of body during stopping along x	$a$	m/s²	—
net force on body during stopping along x	$F_{net}$	N	—
mass of body	$m$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on body	$\| W \|$	N	—
displacement of body during stopping	$\| Δ x \|$	m	—
position of body at end of stopping	$\| x_{1} \|$	m	—
position of body at start of stopping	$\| x_{0} \|$	m	—
average velocity of body during stopping	$\| v_{av} \|$	m/s	—
change of velocity of body during stopping	$\| Δ v \|$	m/s	—
velocity of body at end of stopping from ground	$\| v_{1} \|$	m/s	—
velocity of body at start of stopping from ground	$\| v_{0} \|$	m/s	—
average acceleration of body during stopping	$\| a_{av} \|$	m/s²	—
acceleration of body during stopping	$\| a \|$	m/s²	—
net force on body during stopping	$\| F_{net} \|$	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(stopping, x)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(stopping, x)

c_{av} = \frac{d}{Δ t}

path a time(stopping, x)

Δ v = v_{1} - v_{0}

velocity gained(stopping, x)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(stopping, x)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(stopping, x)

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

position after a time(stopping, x)

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

velocity after a distance(stopping, x)

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

mean of the velocities(stopping, x)

F_{net} = m \cdot a

second law(stopping, x)

| Δ x | = \sqrt{{Δ x}^{2}}

magnitude from components(displacement of body during stopping)

| x_{1} | = \sqrt{{x_{1}}^{2}}

magnitude from components(position of body at end of stopping)

| x_{0} | = \sqrt{{x_{0}}^{2}}

magnitude from components(position of body at start of stopping)

| v_{av} | = \sqrt{{v_{av}}^{2}}

magnitude from components(average velocity of body during stopping)

| Δ v | = \sqrt{{Δ v}^{2}}

magnitude from components(change of velocity of body during stopping)

| v_{1} | = \sqrt{{v_{1}}^{2}}

magnitude from components(velocity of body at end of stopping from ground)

| v_{0} | = \sqrt{{v_{0}}^{2}}

magnitude from components(velocity of body at start of stopping from ground)

| a_{av} | = \sqrt{{a_{av}}^{2}}

magnitude from components(average acceleration of body during stopping)

| a | = \sqrt{a^{2}}

magnitude from components(acceleration of body during stopping)

| F_{net} | = \sqrt{{F_{net}}^{2}}

magnitude from components(net force on body during stopping)

F_{net} = - | F_{net} |

against its motion

m \neq 0

has mass?(body)

F_{net} = 0

free of forces?(body, stopping)