A Bullet Stopped in Wood

solved NewtonsSecondLaw

Q. A bullet of 0.04 kg $m = 0.04$ kg moving at 90 m/s $v_{0} = 90$ m/s enters a heavy wooden block and is stopped $v_{1} = 0$ m/s after a distance of 60 cm $x_{1} = 60$ cm. What is the average resistive force exerted by the block on the bullet, in newtons?

Inputs

mass

m

0.04 kg

speed

v_{0}

90.0 m/s

distance

x_{1}

60 cm

Outputs

force

| F_{net} |

How it's solved

Solved with NewtonsSecondLaw

Given

$m = 0.04$ kg
$v_{0} = 90$ m/s
$x_{1} = 60$ cm
$v_{1} = 0$ m/s from “is stopped”

Assumed

$x_{0} = 0$ m

Find $| F_{net} |$ .

Working

$x_{1} = 60$ cm $= 0.6$ m
velocity after a distance(stopping, x) ${v_{1}}^{2} = {v_{0}}^{2} + 2 \cdot a \cdot (x_{1} - x_{0})$
$0^{2} = 90^{2} + 2 \cdot a \cdot (0.6 - 0)$
$a = −6750$ m/s²
second law(stopping, x) $F_{net} = m \cdot a$
$F_{net} = 0.04 \cdot (−6750)$
$F_{net} = −270$ N
magnitude from components(net force on bullet during stopping) $| F_{net} | = \sqrt{{F_{net}}^{2}}$
$| F_{net} | = \sqrt{{(−270)}^{2}}$
$| F_{net} | = 270$ N

Conclusion

Taking the force as steady, the wood has to undo the bullet's speed over that short distance; the real force varies, so this is its average.

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_bullet_stopped_in_wood do4  kind :numerical5  describes "A bullet of %{mass} moving at %{speed} enters a heavy wooden block and is stopped after a distance " \6            "of %{distance}. %{force}, in newtons?"7  given mass: "0.04 kg"8  given speed: "90 m/s"9  given distance: "60 cm"10  asks force: "What is the average resistive force exerted by the block on the bullet"11end

solution.rb

1require_relative "../questions/a_bullet_stopped_in_wood"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :a_bullet_stopped_in_wood, using: NewtonsSecondLaw do5  objects :bullet6  axes :x7  bullet.goes(:stopping)89  take mass: bullet.mass10  take speed: bullet.initial_velocity11  take distance: bullet.final_position1213  assume bullet.initial_position => "0 m"14  deduce_from "is stopped", bullet.final_velocity => "0 m/s"1516  force = bullet.net_force.magnitude17  answer force:, in: "N"1819  conclusion "Taking the force as steady, the wood has to undo the bullet's speed over that short " \20             "distance; the real force varies, so this is its average."21end

playground.rb

1require_relative "../solutions/a_bullet_stopped_in_wood"23play_with :a_bullet_stopped_in_wood do4  title "A Bullet Stopped in Wood"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 bullet during stopping along x	$Δ x$	m	—
position of bullet at end of stopping along x	$x_{1}$	m	—
position of bullet at start of stopping along x	$x_{0}$	m	—
average velocity of bullet during stopping along x	$v_{av}$	m/s	—
time taken of stopping	$Δ t$	s	0 … 1e+05
average speed of bullet during stopping	$c_{av}$	m/s	—
path length of bullet during stopping	$d$	m	—
change of velocity of bullet during stopping along x	$Δ v$	m/s	—
velocity of bullet at end of stopping from ground along x	$v_{1}$	m/s	—
velocity of bullet at start of stopping from ground along x	$v_{0}$	m/s	—
average acceleration of bullet during stopping along x	$a_{av}$	m/s²	—
acceleration of bullet during stopping along x	$a$	m/s²	—
net force on bullet during stopping along x	$F_{net}$	N	—
mass of bullet	$m$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on bullet	$\| W \|$	N	—
displacement of bullet during stopping	$\| Δ x \|$	m	—
position of bullet at end of stopping	$\| x_{1} \|$	m	—
position of bullet at start of stopping	$\| x_{0} \|$	m	—
average velocity of bullet during stopping	$\| v_{av} \|$	m/s	—
change of velocity of bullet during stopping	$\| Δ v \|$	m/s	—
velocity of bullet at end of stopping from ground	$\| v_{1} \|$	m/s	—
velocity of bullet at start of stopping from ground	$\| v_{0} \|$	m/s	—
average acceleration of bullet during stopping	$\| a_{av} \|$	m/s²	—
acceleration of bullet during stopping	$\| a \|$	m/s²	—
net force on bullet 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 bullet during stopping)

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

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

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

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

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

magnitude from components(average velocity of bullet during stopping)

| Δ v | = \sqrt{{Δ v}^{2}}

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

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

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

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

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

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

magnitude from components(average acceleration of bullet during stopping)

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

magnitude from components(acceleration of bullet during stopping)

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

magnitude from components(net force on bullet during stopping)

m \neq 0

has mass?(bullet)

F_{net} = 0

free of forces?(bullet, stopping)