A Bullet Stopped in Wood

Q. A bullet of 0.04 kg moving at 90 m/s enters a heavy wooden block and is stopped after a distance of 60 cm. What is the average resistive force exerted by the block on the bullet, in newtons? expects 270 N

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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?

The playground

Inputs

mass

m

0.04 kg

speed

v_{0}

90.0 m/s

distance

x_{1}

60 cm

Outputs

force

| F_{net} |

The solution

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 Ruby

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

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

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

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)