Heading at Two Seconds

solved MotionInAPlane

Q. The position of a particle is given by r = 3.0t î $x_{0,x} = 0$ m, $v_{0,x} = 3.0$ m/s, $a_{x} = 0$ m/s² − 2.0t² ĵ $v_{0,y} = 0$ m/s, $a_{y} = -4.0$ m/s² + 4.0 k̂ m, where t is in seconds and the coefficients have the proper units for r to be in metres. What is the direction of the velocity of the particle, measured from the x-axis, at t = 2.0 s $Δ t = 2.0$ s, in degrees?

Inputs

Δ t

2.00 s

Outputs

direction

θ_{1}

How it's solved

Solved with MotionInAPlane

Given

$Δ t = 2.0$ s
$x_{0,x} = 0$ m from “3.0t î”
$v_{0,x} = 3.0$ m/s from “3.0t î”
$a_{x} = 0$ m/s² from “3.0t î”
$v_{0,y} = 0$ m/s from “− 2.0t² ĵ”
$a_{y} = -4.0$ m/s² from “− 2.0t² ĵ”

Find $θ_{1}$ .

Working

velocity after a time(flying, x) $v_{1,x} = v_{0,x} + a_{x} \cdot Δ t$
$v_{1,x} = 3 + 0 \cdot 2$
$v_{1,x} = 3$ m/s
velocity after a time(flying, y) $v_{1,y} = v_{0,y} + a_{y} \cdot Δ t$
$v_{1,y} = 0 + (−4) \cdot 2$
$v_{1,y} = −8$ m/s
magnitude from components(velocity of particle at end of flying from ground) $v_{1} = \sqrt{{v_{1,x}}^{2} + {v_{1,y}}^{2}}$
$v_{1} = \sqrt{3^{2} + {(−8)}^{2}}$
$v_{1} = 8.54400375$ m/s
parts of a velocity(particle, end of flying, x) $v_{1,x} = v_{1} \cdot \cos θ_{1}$
$3 = 8.54400375 \cdot \cos θ_{1}$
$θ_{1} = -69.44°$
$θ_{1} \approx −69.4$ °to 3 significant figures

Conclusion

Along x the particle keeps a steady velocity while along y it speeds up ever faster, so its velocity swings round, further and further below the x-axis.

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 :heading_at_two_seconds do4  kind :numerical5  describes "The position of a particle is given by r = 3.0t î − 2.0t² ĵ + 4.0 k̂ m, where t is in seconds and the " \6            "coefficients have the proper units for r to be in metres. %{direction} at t = %{t}, in degrees?"7  given t: "2.0 s"8  asks direction: "What is the direction of the velocity of the particle, measured from the x-axis,"9end

solution.rb

1require_relative "../questions/heading_at_two_seconds"2require_relative "../science/motion_in_a_plane/motion_in_a_plane"34solve :heading_at_two_seconds, using: MotionInAPlane do5  objects :particle6  axes :x, :y7  particle.goes(:flying)8  take t: flying.time_taken9  deduce_from "3.0t î", particle.initial_position(along: x) => "0 m", particle.initial_velocity(along: x) => "3.0 m/s",10                        particle.acceleration(along: x) => "0 m/s2"11  deduce_from "− 2.0t² ĵ", particle.initial_velocity(along: y) => "0 m/s", particle.acceleration(along: y) => "-4.0 m/s2"12  direction = particle.direction(at: flying.end)13  answer direction:, in: "deg", figures: 314  conclusion "Along x the particle keeps a steady velocity while along y it speeds up ever faster, so its " \15             "velocity swings round, further and further below the x-axis."16end

playground.rb

1require_relative "../solutions/heading_at_two_seconds"23play_with :heading_at_two_seconds do4  title "Heading at Two Seconds"5end

The laws it drew on

quantity	written	units	within	from
volume	$volume$	—	—	Vectors
mass	$mass$	—	—	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	$force$	—	—	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
displacement of particle during flying along x	$Δ x_{x}$	m	—
position of particle at end of flying along x	$x_{1,x}$	m	—
position of particle at start of flying along x	$x_{0,x}$	m	—
displacement of particle during flying along y	$Δ x_{y}$	m	—
position of particle at end of flying along y	$x_{1,y}$	m	—
position of particle at start of flying along y	$x_{0,y}$	m	—
average velocity of particle during flying along x	$v_{av,x}$	m/s	—
time taken of flying	$Δ t$	s	0 … 1e+05
average velocity of particle during flying along y	$v_{av,y}$	m/s	—
average speed of particle during flying	$c_{av}$	m/s	—
path length of particle during flying	$d$	m	—
change of velocity of particle during flying along x	$Δ v_{x}$	m/s	—
velocity of particle at end of flying from ground along x	$v_{1,x}$	m/s	—
velocity of particle at start of flying from ground along x	$v_{0,x}$	m/s	—
change of velocity of particle during flying along y	$Δ v_{y}$	m/s	—
velocity of particle at end of flying from ground along y	$v_{1,y}$	m/s	—
velocity of particle at start of flying from ground along y	$v_{0,y}$	m/s	—
average acceleration of particle during flying along x	$a_{av,x}$	m/s²	—
average acceleration of particle during flying along y	$a_{av,y}$	m/s²	—
acceleration of particle during flying along x	$a_{x}$	m/s²	—
acceleration of particle during flying along y	$a_{y}$	m/s²	—
velocity of ground at start of flying from ground along x	$v_{0,g,x}$	m/s	—
velocity of ground at start of flying from ground	$v_{0,g}$	m/s	—
direction of ground at start of flying from ground	$θ_{0,g}$	—	-3.142 … 3.142
velocity of ground at start of flying from ground along y	$v_{0,g,y}$	m/s	—
velocity of ground at end of flying from ground along x	$v_{1,g,x}$	m/s	—
velocity of ground at end of flying from ground	$v_{1,g}$	m/s	—
direction of ground at end of flying from ground	$θ_{1,g}$	—	-3.142 … 3.142
velocity of ground at end of flying from ground along y	$v_{1,g,y}$	m/s	—
velocity of particle at start of flying from ground	$v_{0}$	m/s	—
direction of particle at start of flying from ground	$θ_{0}$	—	-3.142 … 3.142
velocity of particle at end of flying from ground	$v_{1}$	m/s	—
direction of particle at end of flying from ground	$θ_{1}$	—	-3.142 … 3.142
displacement of particle during flying	$Δ x$	m	—
position of particle at end of flying	$x_{1}$	m	—
position of particle at start of flying	$x_{0}$	m	—
average velocity of particle during flying	$v_{av}$	m/s	—
change of velocity of particle during flying	$Δ v$	m/s	—
average acceleration of particle during flying	$a_{av}$	m/s²	—
acceleration of particle during flying	$a$	m/s²	—

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

v_{o,x} (t) = v_{o} (t) \cdot \cos θ_{o} (t)

parts of a velocity

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} = x_{1,x} - x_{0,x}

change of position(flying, x)

Δ x_{y} = x_{1,y} - x_{0,y}

change of position(flying, y)

v_{av,x} = \frac{Δ x_{x}}{Δ t}

displacement a time(flying, x)

v_{av,y} = \frac{Δ x_{y}}{Δ t}

displacement a time(flying, y)

c_{av} = \frac{d}{Δ t}

path a time(flying, x)

c_{av} = \frac{d}{Δ t}

path a time(flying, y)

Δ v_{x} = v_{1,x} - v_{0,x}

velocity gained(flying, x)

Δ v_{y} = v_{1,y} - v_{0,y}

velocity gained(flying, y)

a_{av,x} = \frac{Δ v_{x}}{Δ t}

velocity gained a time(flying, x)

a_{av,y} = \frac{Δ v_{y}}{Δ t}

velocity gained a time(flying, y)

v_{1,x} = v_{0,x} + a_{x} \cdot Δ t

velocity after a time(flying, x)

v_{1,y} = v_{0,y} + a_{y} \cdot Δ t

velocity after a time(flying, y)

x_{1,x} = x_{0,x} + v_{0,x} \cdot Δ t + \frac{a_{x} \cdot {Δ t}^{2}}{2}

position after a time(flying, x)

x_{1,y} = x_{0,y} + v_{0,y} \cdot Δ t + \frac{a_{y} \cdot {Δ t}^{2}}{2}

position after a time(flying, y)

{v_{1,x}}^{2} = {v_{0,x}}^{2} + 2 \cdot a_{x} \cdot (x_{1,x} - x_{0,x})

velocity after a distance(flying, x)

{v_{1,y}}^{2} = {v_{0,y}}^{2} + 2 \cdot a_{y} \cdot (x_{1,y} - x_{0,y})

velocity after a distance(flying, y)

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

mean of the velocities(flying, x)

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

mean of the velocities(flying, y)

v_{0,g,x} = v_{0,g} \cdot \cos θ_{0,g}

parts of a velocity(ground, start of flying, x)

v_{0,g,y} = v_{0,g} \cdot \sin θ_{0,g}

parts of a velocity(ground, start of flying, y)

v_{1,g,x} = v_{1,g} \cdot \cos θ_{1,g}

parts of a velocity(ground, end of flying, x)

v_{1,g,y} = v_{1,g} \cdot \sin θ_{1,g}

parts of a velocity(ground, end of flying, y)

v_{0,x} = v_{0} \cdot \cos θ_{0}

parts of a velocity(particle, start of flying, x)

v_{0,y} = v_{0} \cdot \sin θ_{0}

parts of a velocity(particle, start of flying, y)

v_{1,x} = v_{1} \cdot \cos θ_{1}

parts of a velocity(particle, end of flying, x)

v_{1,y} = v_{1} \cdot \sin θ_{1}

parts of a velocity(particle, end of flying, y)

Δ x = \sqrt{{Δ x_{x}}^{2} + {Δ x_{y}}^{2}}

magnitude from components(displacement of particle during flying)

x_{1} = \sqrt{{x_{1,x}}^{2} + {x_{1,y}}^{2}}

magnitude from components(position of particle at end of flying)

x_{0} = \sqrt{{x_{0,x}}^{2} + {x_{0,y}}^{2}}

magnitude from components(position of particle at start of flying)

v_{av} = \sqrt{{v_{av,x}}^{2} + {v_{av,y}}^{2}}

magnitude from components(average velocity of particle during flying)

Δ v = \sqrt{{Δ v_{x}}^{2} + {Δ v_{y}}^{2}}

magnitude from components(change of velocity of particle during flying)

v_{1} = \sqrt{{v_{1,x}}^{2} + {v_{1,y}}^{2}}

magnitude from components(velocity of particle at end of flying from ground)

v_{0} = \sqrt{{v_{0,x}}^{2} + {v_{0,y}}^{2}}

magnitude from components(velocity of particle at start of flying from ground)

a_{av} = \sqrt{{a_{av,x}}^{2} + {a_{av,y}}^{2}}

magnitude from components(average acceleration of particle during flying)

a = \sqrt{{a_{x}}^{2} + {a_{y}}^{2}}

magnitude from components(acceleration of particle during flying)

v_{0,g} = \sqrt{{v_{0,g,x}}^{2} + {v_{0,g,y}}^{2}}

magnitude from components(velocity of ground at start of flying from ground)

v_{1,g} = \sqrt{{v_{1,g,x}}^{2} + {v_{1,g,y}}^{2}}

magnitude from components(velocity of ground at end of flying from ground)