Pushed Two Ways — phryzby

Q. A body of 5 kg is pushed by two forces at right angles to each other, 8 N and 6 N. How strongly does it accelerate, in m/s²? expects 2 m/s²

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The equations it used

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Q. A body of 5 kg $m = 5$ kg is pushed by two forces at right angles $θ_{f;s} = 90 °$ to each other, 8 N $F_{f} = 8$ N and 6 N $F_{s} = 6$ N. How strongly does it accelerate, in m/s²?

The playground

Inputs

mass

m

5 kg

first force

F_{f}

8 N

second force

F_{s}

6 N

Outputs

acceleration

a

The solution

Solved with NewtonsSecondLaw

Given

$m = 5$ kg
$F_{f} = 8$ N
$F_{s} = 6$ N
$θ_{f;s} = 90 °$ from “at right angles”
$F_{net}$ : the positive one of the values that fit from “How strongly”

Find $a$ .

Working

net force is the resultant(body, pushed) ${F_{net}}^{2} = {F_{f}}^{2} + {F_{s}}^{2} + 2 \cdot F_{f} \cdot F_{s} \cdot \cos θ_{f;s}$
${F_{net}}^{2} = 8^{2} + 6^{2} + 2 \cdot 8 \cdot 6 \cdot \cos 90 °$
$F_{net} = −10, 10$ Nthe positive one is taken
$F_{net} = 10$ N
second law(pushed) $F_{net} = m \cdot a$
$10 = 5 \cdot a$
$a = 2$ m/s²

Conclusion

At right angles the two forces add like the sides of a right triangle, and the body accelerates along the hypotenuse at that force over its mass.

The Ruby

1require_relative "../physics"23question :pushed_two_ways do4  kind :numerical5  describes "A body of %{mass} is pushed by two forces at right angles to each other, %{first_force} and " \6            "%{second_force}. %{acceleration}, in m/s²?"7  given mass: "5 kg"8  given first_force: "8 N"9  given second_force: "6 N"10  asks acceleration: "How strongly does it accelerate"11end

1require_relative "../questions/pushed_two_ways"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :pushed_two_ways, using: NewtonsSecondLaw do5  objects :body, :first_pusher, :second_pusher6  body.goes(:pushed)78  take mass: body.mass9  take first_force: body.force(by: first_pusher).magnitude10  take second_force: body.force(by: second_pusher).magnitude1112  deduce_from "at right angles", first_force.perpendicular_to(second_force)13  deduce_from "How strongly", body.net_force => :positive1415  acceleration = body.acceleration.magnitude16  answer acceleration:, in: "m/s2"1718  conclusion "At right angles the two forces add like the sides of a right triangle, and the body " \19             "accelerates along the hypotenuse at that force over its mass."20end

1require_relative "../solutions/pushed_two_ways"23play_with :pushed_two_ways do4  title "Pushed Two Ways"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
angle between force on body by first pusher during pushed and force on body by second pusher during pushed	$θ_{f;s}$	—	0 … 3.142
displacement of body during pushed	$Δ x$	m	—
position of body at end of pushed	$x_{1}$	m	—
position of body at start of pushed	$x_{0}$	m	—
average velocity of body during pushed	$v_{av}$	m/s	—
time taken of pushed	$Δ t$	s	0 … 1e+05
average speed of body during pushed	$c_{av}$	m/s	—
path length of body during pushed	$d$	m	—
change of velocity of body during pushed	$Δ v$	m/s	—
velocity of body at end of pushed from ground	$v_{1}$	m/s	—
velocity of body at start of pushed from ground	$v_{0}$	m/s	—
average acceleration of body during pushed	$a_{av}$	m/s²	—
acceleration of body during pushed	$a$	m/s²	—
net force on body during pushed	$F_{net}$	N	—
mass of body	$m$	kg	0 … ∞
mass of first pusher	$m_{f}$	kg	0 … ∞
mass of second pusher	$m_{s}$	kg	0 … ∞
weight on ground	$W_{g}$	N	—
weight on body	$W$	N	—
weight on first pusher	$W_{f}$	N	—
weight on second pusher	$W_{s}$	N	—
force on body by first pusher during pushed	$F_{f}$	N	—
force on body by second pusher during pushed	$F_{s}$	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(pushed)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(pushed)

c_{av} = \frac{d}{Δ t}

path a time(pushed)

Δ v = v_{1} - v_{0}

velocity gained(pushed)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(pushed)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(pushed)

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

position after a time(pushed)

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

velocity after a distance(pushed)

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

mean of the velocities(pushed)

F_{net} = m \cdot a

second law(pushed)

{F_{net}}^{2} = {F_{f}}^{2} + {F_{s}}^{2} + 2 \cdot F_{f} \cdot F_{s} \cdot \cos θ_{f;s}

net force is the resultant(body, pushed)

m \neq 0

has mass?(body)

m_{f} \neq 0

has mass?(first pusher)

m_{s} \neq 0

has mass?(second pusher)

F_{net} = 0

free of forces?(body, pushed)