A Pebble Thrown Up

solved NewtonsSecondLaw

Q. A pebble of mass 0.05 kg $m = 0.05$ kg is thrown vertically upwards. Ignore air resistance $F_{a} = 0$ N, and take g as $W = - | W |$ 10 m/s² $g = 10$ m/s². What is the magnitude of the net force on the pebble during its upward motion, in newtons?

Inputs

mass

m

0.05 kg

gravity

g

10 m/s²

Outputs

force

| F_{net} |

How it's solved

Solved with NewtonsSecondLaw

Given

$m = 0.05$ kg
$g = 10$ m/s²
$F_{a} = 0$ N from “Ignore air resistance”
$W = - | W |$ from “take g as”

Find $| F_{net} |$ .

Working

weight of a body(pebble) $| W | = m \cdot g$
$| W | = 0.05 \cdot 10$
$| W | = 0.5$ N
take g as $W = - | W |$
$W = - 0.5$
$W = −0.5$ N
net force is the resultant(pebble, rising, y) $F_{net} = F_{a} + W$
$F_{net} = 0 + (−0.5)$
$F_{net} = −0.5$ N
magnitude from components(net force on pebble during rising) $| F_{net} | = \sqrt{{F_{net}}^{2}}$
$| F_{net} | = \sqrt{{(−0.5)}^{2}}$
$| F_{net} | = 0.5$ N

Conclusion

With the air left out, the only force on the pebble is its weight, downward, whether it is going up, coming down or still for an instant at the top.

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_pebble_thrown_up do4  kind :numerical5  describes "A pebble of mass %{mass} is thrown vertically upwards. Ignore air resistance, and take g as %{gravity}. " \6            "%{force} during its upward motion, in newtons?"7  given mass: "0.05 kg"8  given gravity: "10 m/s2"9  asks force: "What is the magnitude of the net force on the pebble"10end

solution.rb

1require_relative "../questions/a_pebble_thrown_up"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :a_pebble_thrown_up, using: NewtonsSecondLaw do5  objects :pebble, :air6  axes :y7  pebble.goes(:rising)8  deduce_from "take g as", pebble.weight.direction => -y910  take mass: pebble.mass11  take gravity: g1213  deduce_from "Ignore air resistance", pebble.force(by: air) => "0 N"1415  force = pebble.net_force.magnitude16  answer force:, in: "N"1718  conclusion "With the air left out, the only force on the pebble is its weight, downward, whether it is " \19             "going up, coming down or still for an instant at the top."20end

playground.rb

1require_relative "../solutions/a_pebble_thrown_up"23play_with :a_pebble_thrown_up do4  title "A Pebble Thrown Up"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 pebble during rising along y	$Δ x$	m	—
position of pebble at end of rising along y	$x_{1}$	m	—
position of pebble at start of rising along y	$x_{0}$	m	—
average velocity of pebble during rising along y	$v_{av}$	m/s	—
time taken of rising	$Δ t$	s	0 … 1e+05
average speed of pebble during rising	$c_{av}$	m/s	—
path length of pebble during rising	$d$	m	—
change of velocity of pebble during rising along y	$Δ v$	m/s	—
velocity of pebble at end of rising from ground along y	$v_{1}$	m/s	—
velocity of pebble at start of rising from ground along y	$v_{0}$	m/s	—
average acceleration of pebble during rising along y	$a_{av}$	m/s²	—
acceleration of pebble during rising along y	$a$	m/s²	—
net force on pebble during rising along y	$F_{net}$	N	—
mass of pebble	$m$	kg	0 … ∞
mass of air	$m_{a}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on pebble	$\| W \|$	N	—
weight on air	$\| W_{a} \|$	N	—
force on pebble by air during rising	$\| F_{a} \|$	N	—
weight on pebble along y	$W$	N	—
force on pebble by air during rising along y	$F_{a}$	N	—
displacement of pebble during rising	$\| Δ x \|$	m	—
position of pebble at end of rising	$\| x_{1} \|$	m	—
position of pebble at start of rising	$\| x_{0} \|$	m	—
average velocity of pebble during rising	$\| v_{av} \|$	m/s	—
change of velocity of pebble during rising	$\| Δ v \|$	m/s	—
velocity of pebble at end of rising from ground	$\| v_{1} \|$	m/s	—
velocity of pebble at start of rising from ground	$\| v_{0} \|$	m/s	—
average acceleration of pebble during rising	$\| a_{av} \|$	m/s²	—
acceleration of pebble during rising	$\| a \|$	m/s²	—
net force on pebble during rising	$\| F_{net} \|$	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(rising, y)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(rising, y)

c_{av} = \frac{d}{Δ t}

path a time(rising, y)

Δ v = v_{1} - v_{0}

velocity gained(rising, y)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(rising, y)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(rising, y)

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

position after a time(rising, y)

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

velocity after a distance(rising, y)

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

mean of the velocities(rising, y)

F_{net} = m \cdot a

second law(rising, y)

| W | = m \cdot g

weight of a body(pebble)

F_{net} = F_{a} + W

net force is the resultant(pebble, rising, y)

| Δ x | = \sqrt{{Δ x}^{2}}

magnitude from components(displacement of pebble during rising)

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

magnitude from components(position of pebble at end of rising)

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

magnitude from components(position of pebble at start of rising)

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

magnitude from components(average velocity of pebble during rising)

| Δ v | = \sqrt{{Δ v}^{2}}

magnitude from components(change of velocity of pebble during rising)

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

magnitude from components(velocity of pebble at end of rising from ground)

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

magnitude from components(velocity of pebble at start of rising from ground)

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

magnitude from components(average acceleration of pebble during rising)

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

magnitude from components(acceleration of pebble during rising)

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

magnitude from components(net force on pebble during rising)

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

magnitude from components(weight on pebble)

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

magnitude from components(force on pebble by air during rising)

W = - | W |

take g as

m \neq 0

has mass?(pebble)

m_{a} \neq 0

has mass?(air)

F_{net} = 0

free of forces?(pebble, rising)