Heading at Two Seconds

Q. 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 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, in degrees? expects -69.4°

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The web of concepts

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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?

The playground

Inputs

Δ t

2.00 s

Outputs

direction

θ_{1}

The solution

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 Ruby

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

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

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

The laws

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)