Two Billiard Balls Rebound

Q. Two billiard balls each of mass 0.05 kg moving in opposite directions with speed 6 m/s collide and rebound with the same speed. What is the impulse imparted to each ball due to the other, in N s? expects 0.6 N s

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Q. Two billiard balls each of mass 0.05 kg $m = 0.05$ kg moving in opposite directions with speed 6 m/s $v_{0} = 6$ m/s collide and rebound with the same speed $v_{1} = - v_{0}$ . What is the impulse imparted to each ball due to the other, in N s?

The playground

Inputs

mass

m

0.05 kg

speed

v_{0}

6.0 m/s

Outputs

impulse

| J |

The solution

Solved with Momentum

Given

$m = 0.05$ kg
$v_{0} = 6$ m/s
$v_{1} = - v_{0}$ from “rebound with the same speed”

Find $| J |$ .

Working

rebound with the same speed $v_{1} = - v_{0}$
$v_{1} = - 6$
$v_{1} = −6$ m/s
momentum at the end(colliding, x) $p_{1} = m \cdot v_{1}$
$p_{1} = 0.05 \cdot (−6)$
$p_{1} = −0.3$ kg m/s
momentum at the start(colliding, x) $p_{0} = m \cdot v_{0}$
$p_{0} = 0.05 \cdot 6$
$p_{0} = 0.3$ kg m/s
impulse is the change of momentum(colliding, x) $J = p_{1} - p_{0}$
$J = (−0.3) - 0.3$
$J = −0.6$ kg m/s
magnitude from components(impulse on ball during colliding) $| J | = \sqrt{J^{2}}$
$| J | = \sqrt{{(−0.6)}^{2}}$
$| J | = 0.6$ N s
$| J | = 0.6$ kg m/s $= 0.6$ N s

Conclusion

Each ball is turned right round, so the other must take away all its momentum and give it as much again the other way; the two impulses are equal and opposite.

The Ruby

1require_relative "../physics"23question :two_billiard_balls_rebound do4  kind :numerical5  describes "Two billiard balls each of mass %{mass} moving in opposite directions with speed %{speed} collide and " \6            "rebound with the same speed. %{impulse}, in N s?"7  given mass: "0.05 kg"8  given speed: "6 m/s"9  asks impulse: "What is the impulse imparted to each ball due to the other"10end

1require_relative "../questions/two_billiard_balls_rebound"2require_relative "../science/laws_of_motion/momentum"34solve :two_billiard_balls_rebound, using: Momentum do5  objects :ball6  axes :x7  ball.goes(:colliding)89  take mass: ball.mass10  take speed: ball.initial_velocity1112  deduce_from "rebound with the same speed", ball.final_velocity => -ball.initial_velocity1314  impulse = ball.impulse.magnitude15  answer impulse:, in: "N s"1617  conclusion "Each ball is turned right round, so the other must take away all its momentum and give it as " \18             "much again the other way; the two impulses are equal and opposite."19end

1require_relative "../solutions/two_billiard_balls_rebound"23play_with :two_billiard_balls_rebound do4  title "Two Billiard Balls Rebound"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	$J$	kg m/s	—	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
momentum	$p$	kg m/s	—	Momentum
starting momentum	$p_{0}$	kg m/s	—	Momentum
final momentum	$p$	kg m/s	—	Momentum
displacement of ball during colliding along x	$Δ x$	m	—
position of ball at end of colliding along x	$x_{1}$	m	—
position of ball at start of colliding along x	$x_{0}$	m	—
average velocity of ball during colliding along x	$v_{av}$	m/s	—
time taken of colliding	$Δ t$	s	0 … 1e+05
average speed of ball during colliding	$c_{av}$	m/s	—
path length of ball during colliding	$d$	m	—
change of velocity of ball during colliding along x	$Δ v$	m/s	—
velocity of ball at end of colliding from ground along x	$v_{1}$	m/s	—
velocity of ball at start of colliding from ground along x	$v_{0}$	m/s	—
average acceleration of ball during colliding along x	$a_{av}$	m/s²	—
acceleration of ball during colliding along x	$a$	m/s²	—
net force on ball during colliding along x	$F_{net}$	N	—
mass of ball	$m$	kg	0 … ∞
momentum of ball at start of colliding along x	$p_{0}$	kg m/s	—
momentum of ball at end of colliding along x	$p_{1}$	kg m/s	—
impulse on ball during colliding along x	$J$	kg m/s	—
weight on ground	$\| W_{g} \|$	N	—
weight on ball	$\| W \|$	N	—
displacement of ball during colliding	$\| Δ x \|$	m	—
position of ball at end of colliding	$\| x_{1} \|$	m	—
position of ball at start of colliding	$\| x_{0} \|$	m	—
average velocity of ball during colliding	$\| v_{av} \|$	m/s	—
change of velocity of ball during colliding	$\| Δ v \|$	m/s	—
velocity of ball at end of colliding from ground	$\| v_{1} \|$	m/s	—
velocity of ball at start of colliding from ground	$\| v_{0} \|$	m/s	—
average acceleration of ball during colliding	$\| a_{av} \|$	m/s²	—
acceleration of ball during colliding	$\| a \|$	m/s²	—
net force on ball during colliding	$\| F_{net} \|$	N	—
momentum of ball at start of colliding	$\| p_{0} \|$	kg m/s	—
momentum of ball at end of colliding	$\| p_{1} \|$	kg m/s	—
impulse on ball during colliding	$\| J \|$	kg m/s	—

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

p_{0} \equiv p_{i} (start of motion)

starting momentum

p \equiv p_{i} (end of motion)

final momentum

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(colliding, x)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(colliding, x)

c_{av} = \frac{d}{Δ t}

path a time(colliding, x)

Δ v = v_{1} - v_{0}

velocity gained(colliding, x)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(colliding, x)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(colliding, x)

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

position after a time(colliding, x)

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

velocity after a distance(colliding, x)

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

mean of the velocities(colliding, x)

F_{net} = m \cdot a

second law(colliding, x)

p_{0} = m \cdot v_{0}

momentum at the start(colliding, x)

p_{1} = m \cdot v_{1}

momentum at the end(colliding, x)

J = p_{1} - p_{0}

impulse is the change of momentum(colliding, x)

J = F_{net} \cdot Δ t

impulse is force over time(colliding, x)

| Δ x | = \sqrt{{Δ x}^{2}}

magnitude from components(displacement of ball during colliding)

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

magnitude from components(position of ball at end of colliding)

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

magnitude from components(position of ball at start of colliding)

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

magnitude from components(average velocity of ball during colliding)

| Δ v | = \sqrt{{Δ v}^{2}}

magnitude from components(change of velocity of ball during colliding)

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

magnitude from components(velocity of ball at end of colliding from ground)

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

magnitude from components(velocity of ball at start of colliding from ground)

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

magnitude from components(average acceleration of ball during colliding)

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

magnitude from components(acceleration of ball during colliding)

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

magnitude from components(net force on ball during colliding)

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

magnitude from components(momentum of ball at start of colliding)

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

magnitude from components(momentum of ball at end of colliding)

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

magnitude from components(impulse on ball during colliding)

v_{1} = - v_{0}

rebound with the same speed

m \neq 0

has mass?(ball)

F_{net} = 0

free of forces?(ball, colliding)