Dropped From a Truck

solved MotionInAPlane

Q. A truck starts from rest $v_{x} (start) = 0$ m/s, $v_{y} (start) = 0$ m/s and accelerates uniformly at 2.0 m/s² $a_{x} (c) = 2.0$ m/s². At t = 10 s $Δ t_{c} = 10$ s, a stone is dropped by a person standing on the top of the truck $a_{y} (c) = 0$ m/s² (6 m $x_{y} (dropped) = 6$ m high from the ground). Neglect air resistance $a_{x} (f) = 0$ m/s², $a_{y} (f) = - g$ , and take g as 10 m/s² $g = 10$ m/s². What is the speed of the stone at t = 11 s $Δ t_{a} = 11$ s, in m/s?

Inputs

acceleration

a_{x}

2 m/s²

dropped at

Δ t_{c}

10.00 s

height

x_{y}

6.00 m

gravity

g

10 m/s²

asked at

Δ t_{a}

11.00 s

Outputs

speed

v

How it's solved

Solved with MotionInAPlane

Given

$a_{x} (c) = 2.0$ m/s²
$Δ t_{c} = 10$ s
$x_{y} (dropped) = 6$ m
$g = 10$ m/s²
$Δ t_{a} = 11$ s
$v_{x} (start) = 0$ m/s from “starts from rest”
$v_{y} (start) = 0$ m/s from “starts from rest”
$a_{y} (c) = 0$ m/s² from “standing on the top of the truck”
$a_{x} (f) = 0$ m/s² from “Neglect air resistance”
$a_{y} (f) = - g$ from “Neglect air resistance”

Find $v (later)$ .

Working

velocity after a time(carried, x) $v_{x} (dropped) = v_{x} (start) + a_{x} (c) \cdot Δ t_{c}$
$v_{x} (dropped) = 0 + 2 \cdot 10$
$v_{x} (dropped) = 20$ m/s
time in parts(all along) $Δ t_{a} = Δ t_{c} + Δ t_{f}$
$11 = 10 + Δ t_{f}$
$Δ t_{f} = 1$ s
velocity after a time(falling, x) $v_{x} (later) = v_{x} (dropped) + a_{x} (f) \cdot Δ t_{f}$
$v_{x} (later) = 20 + 0 \cdot 1$
$v_{x} (later) = 20$ m/s
velocity after a time(carried, y) $v_{y} (dropped) = v_{y} (start) + a_{y} (c) \cdot Δ t_{c}$
$v_{y} (dropped) = 0 + 0 \cdot 10$
$v_{y} (dropped) = 0$ m/s
Neglect air resistance $a_{y} (f) = - g$
$a_{y} (f) = - 10$
$a_{y} (f) = −10$ m/s²
velocity after a time(falling, y) $v_{y} (later) = v_{y} (dropped) + a_{y} (f) \cdot Δ t_{f}$
$v_{y} (later) = 0 + (−10) \cdot 1$
$v_{y} (later) = −10$ m/s
magnitude from components(velocity of stone at later from ground) $v (later) = \sqrt{{v_{x} (later)}^{2} + {v_{y} (later)}^{2}}$
$v (later) = \sqrt{20^{2} + {(−10)}^{2}}$
$v (later) = 22.4 m/s$
$v (later) \approx 22.4$ m/sto 3 significant figures

Conclusion

The stone leaves the truck with the truck's velocity and keeps it sideways, since nothing pushes it along any more, while gravity alone speeds it downward.

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 :dropped_from_a_truck do4  kind :numerical5  describes "A truck starts from rest and accelerates uniformly at %{acceleration}. At t = %{dropped_at}, a stone is " \6            "dropped by a person standing on the top of the truck (%{height} high from the ground). Neglect air " \7            "resistance, and take g as %{gravity}. %{speed} at t = %{asked_at}, in m/s?"8  given acceleration: "2.0 m/s2"9  given dropped_at: "10 s"10  given height: "6 m"11  given gravity: "10 m/s2"12  given asked_at: "11 s"13  asks speed: "What is the speed of the stone"14end

solution.rb

1require_relative "../questions/dropped_from_a_truck"2require_relative "../science/motion_in_a_plane/motion_in_a_plane"34solve :dropped_from_a_truck, using: MotionInAPlane do5  objects :stone6  axes :x, :y7  instants :start, :dropped, :later8  stone.goes(:carried, during: start..dropped)9  stone.goes(:falling, during: dropped..later)10  stone.goes(:all_along, during: start..later)1112  take acceleration: stone.acceleration(during: carried, along: x)13  take dropped_at: carried.time_taken14  take height: stone.position(at: dropped, along: y)15  take gravity: g16  take asked_at: all_along.time_taken1718  deduce_from "starts from rest", stone.velocity(at: start, along: x) => "0 m/s", stone.velocity(at: start, along: y) => "0 m/s"19  deduce_from "standing on the top of the truck", stone.acceleration(during: carried, along: y) => "0 m/s2"20  deduce_from "Neglect air resistance", stone.acceleration(during: falling, along: x) => "0 m/s2",21                                        stone.acceleration(during: falling, along: y) => -g2223  speed = stone.speed(at: later)24  answer speed:, in: "m/s", figures: 32526  conclusion "The stone leaves the truck with the truck's velocity and keeps it sideways, since nothing pushes " \27             "it along any more, while gravity alone speeds it downward."28end

playground.rb

1require_relative "../solutions/dropped_from_a_truck"23play_with :dropped_from_a_truck do4  title "Dropped From a Truck"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 stone during carried along x	$Δ x_{x} (c)$	m	—
position of stone at dropped along x	$x_{x} (dropped)$	m	—
position of stone at start along x	$x_{x} (start)$	m	—
displacement of stone during carried along y	$Δ x_{y} (c)$	m	—
position of stone at dropped along y	$x_{y} (dropped)$	m	—
position of stone at start along y	$x_{y} (start)$	m	—
displacement of stone during falling along x	$Δ x_{x} (f)$	m	—
position of stone at later along x	$x_{x} (later)$	m	—
displacement of stone during falling along y	$Δ x_{y} (f)$	m	—
position of stone at later along y	$x_{y} (later)$	m	—
displacement of stone during all along along x	$Δ x_{x} (a)$	m	—
displacement of stone during all along along y	$Δ x_{y} (a)$	m	—
average velocity of stone during carried along x	$v_{av,x} (c)$	m/s	—
time taken of carried	$Δ t_{c}$	s	0 … 1e+05
average velocity of stone during carried along y	$v_{av,y} (c)$	m/s	—
average velocity of stone during falling along x	$v_{av,x} (f)$	m/s	—
time taken of falling	$Δ t_{f}$	s	0 … 1e+05
average velocity of stone during falling along y	$v_{av,y} (f)$	m/s	—
average velocity of stone during all along along x	$v_{av,x} (a)$	m/s	—
time taken of all along	$Δ t_{a}$	s	0 … 1e+05
average velocity of stone during all along along y	$v_{av,y} (a)$	m/s	—
average speed of stone during carried	$c_{av} (c)$	m/s	—
path length of stone during carried	$d (c)$	m	—
average speed of stone during falling	$c_{av} (f)$	m/s	—
path length of stone during falling	$d (f)$	m	—
average speed of stone during all along	$c_{av} (a)$	m/s	—
path length of stone during all along	$d (a)$	m	—
change of velocity of stone during carried along x	$Δ v_{x} (c)$	m/s	—
velocity of stone at dropped from ground along x	$v_{x} (dropped)$	m/s	—
velocity of stone at start from ground along x	$v_{x} (start)$	m/s	—
change of velocity of stone during carried along y	$Δ v_{y} (c)$	m/s	—
velocity of stone at dropped from ground along y	$v_{y} (dropped)$	m/s	—
velocity of stone at start from ground along y	$v_{y} (start)$	m/s	—
change of velocity of stone during falling along x	$Δ v_{x} (f)$	m/s	—
velocity of stone at later from ground along x	$v_{x} (later)$	m/s	—
change of velocity of stone during falling along y	$Δ v_{y} (f)$	m/s	—
velocity of stone at later from ground along y	$v_{y} (later)$	m/s	—
change of velocity of stone during all along along x	$Δ v_{x} (a)$	m/s	—
change of velocity of stone during all along along y	$Δ v_{y} (a)$	m/s	—
average acceleration of stone during carried along x	$a_{av,x} (c)$	m/s²	—
average acceleration of stone during carried along y	$a_{av,y} (c)$	m/s²	—
average acceleration of stone during falling along x	$a_{av,x} (f)$	m/s²	—
average acceleration of stone during falling along y	$a_{av,y} (f)$	m/s²	—
average acceleration of stone during all along along x	$a_{av,x} (a)$	m/s²	—
average acceleration of stone during all along along y	$a_{av,y} (a)$	m/s²	—
acceleration of stone during carried along x	$a_{x} (c)$	m/s²	—
acceleration of stone during carried along y	$a_{y} (c)$	m/s²	—
acceleration of stone during falling along x	$a_{x} (f)$	m/s²	—
acceleration of stone during falling along y	$a_{y} (f)$	m/s²	—
acceleration of stone during all along along x	$a_{x} (a)$	m/s²	—
acceleration of stone during all along along y	$a_{y} (a)$	m/s²	—
acceleration of stone during carried	$a (c)$	m/s²	—
acceleration of stone during all along	$a (a)$	m/s²	—
acceleration of stone during falling	$a (f)$	m/s²	—
velocity of ground at start from ground along x	$v_{g,x} (start)$	m/s	—
velocity of ground at start from ground	$v_{g} (start)$	m/s	—
direction of ground at start from ground	$θ_{g} (start)$	—	-3.142 … 3.142
velocity of ground at start from ground along y	$v_{g,y} (start)$	m/s	—
velocity of ground at dropped from ground along x	$v_{g,x} (dropped)$	m/s	—
velocity of ground at dropped from ground	$v_{g} (dropped)$	m/s	—
direction of ground at dropped from ground	$θ_{g} (dropped)$	—	-3.142 … 3.142
velocity of ground at dropped from ground along y	$v_{g,y} (dropped)$	m/s	—
velocity of ground at later from ground along x	$v_{g,x} (later)$	m/s	—
velocity of ground at later from ground	$v_{g} (later)$	m/s	—
direction of ground at later from ground	$θ_{g} (later)$	—	-3.142 … 3.142
velocity of ground at later from ground along y	$v_{g,y} (later)$	m/s	—
velocity of stone at start from ground	$v (start)$	m/s	—
direction of stone at start from ground	$θ (start)$	—	-3.142 … 3.142
velocity of stone at dropped from ground	$v (dropped)$	m/s	—
direction of stone at dropped from ground	$θ (dropped)$	—	-3.142 … 3.142
velocity of stone at later from ground	$v (later)$	m/s	—
direction of stone at later from ground	$θ (later)$	—	-3.142 … 3.142
displacement of stone during carried	$Δ x (c)$	m	—
position of stone at dropped	$x (dropped)$	m	—
position of stone at start	$x (start)$	m	—
displacement of stone during falling	$Δ x (f)$	m	—
position of stone at later	$x (later)$	m	—
displacement of stone during all along	$Δ x (a)$	m	—
average velocity of stone during carried	$v_{av} (c)$	m/s	—
average velocity of stone during falling	$v_{av} (f)$	m/s	—
average velocity of stone during all along	$v_{av} (a)$	m/s	—
change of velocity of stone during carried	$Δ v (c)$	m/s	—
change of velocity of stone during falling	$Δ v (f)$	m/s	—
change of velocity of stone during all along	$Δ v (a)$	m/s	—
average acceleration of stone during carried	$a_{av} (c)$	m/s²	—
average acceleration of stone during falling	$a_{av} (f)$	m/s²	—
average acceleration of stone during all along	$a_{av} (a)$	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} (c) = x_{x} (dropped) - x_{x} (start)

change of position(carried, x)

Δ x_{y} (c) = x_{y} (dropped) - x_{y} (start)

change of position(carried, y)

Δ x_{x} (f) = x_{x} (later) - x_{x} (dropped)

change of position(falling, x)

Δ x_{y} (f) = x_{y} (later) - x_{y} (dropped)

change of position(falling, y)

Δ x_{x} (a) = x_{x} (later) - x_{x} (start)

change of position(all along, x)

Δ x_{y} (a) = x_{y} (later) - x_{y} (start)

change of position(all along, y)

v_{av,x} (c) = \frac{Δ x_{x} (c)}{Δ t_{c}}

displacement a time(carried, x)

v_{av,y} (c) = \frac{Δ x_{y} (c)}{Δ t_{c}}

displacement a time(carried, y)

v_{av,x} (f) = \frac{Δ x_{x} (f)}{Δ t_{f}}

displacement a time(falling, x)

v_{av,y} (f) = \frac{Δ x_{y} (f)}{Δ t_{f}}

displacement a time(falling, y)

v_{av,x} (a) = \frac{Δ x_{x} (a)}{Δ t_{a}}

displacement a time(all along, x)

v_{av,y} (a) = \frac{Δ x_{y} (a)}{Δ t_{a}}

displacement a time(all along, y)

c_{av} (c) = \frac{d (c)}{Δ t_{c}}

path a time(carried, x)

c_{av} (c) = \frac{d (c)}{Δ t_{c}}

path a time(carried, y)

c_{av} (f) = \frac{d (f)}{Δ t_{f}}

path a time(falling, x)

c_{av} (f) = \frac{d (f)}{Δ t_{f}}

path a time(falling, y)

c_{av} (a) = \frac{d (a)}{Δ t_{a}}

path a time(all along, x)

c_{av} (a) = \frac{d (a)}{Δ t_{a}}

path a time(all along, y)

Δ v_{x} (c) = v_{x} (dropped) - v_{x} (start)

velocity gained(carried, x)

Δ v_{y} (c) = v_{y} (dropped) - v_{y} (start)

velocity gained(carried, y)

Δ v_{x} (f) = v_{x} (later) - v_{x} (dropped)

velocity gained(falling, x)

Δ v_{y} (f) = v_{y} (later) - v_{y} (dropped)

velocity gained(falling, y)

Δ v_{x} (a) = v_{x} (later) - v_{x} (start)

velocity gained(all along, x)

Δ v_{y} (a) = v_{y} (later) - v_{y} (start)

velocity gained(all along, y)

a_{av,x} (c) = \frac{Δ v_{x} (c)}{Δ t_{c}}

velocity gained a time(carried, x)

a_{av,y} (c) = \frac{Δ v_{y} (c)}{Δ t_{c}}

velocity gained a time(carried, y)

a_{av,x} (f) = \frac{Δ v_{x} (f)}{Δ t_{f}}

velocity gained a time(falling, x)

a_{av,y} (f) = \frac{Δ v_{y} (f)}{Δ t_{f}}

velocity gained a time(falling, y)

a_{av,x} (a) = \frac{Δ v_{x} (a)}{Δ t_{a}}

velocity gained a time(all along, x)

a_{av,y} (a) = \frac{Δ v_{y} (a)}{Δ t_{a}}

velocity gained a time(all along, y)

v_{x} (dropped) = v_{x} (start) + a_{x} (c) \cdot Δ t_{c}

velocity after a time(carried, x)

v_{y} (dropped) = v_{y} (start) + a_{y} (c) \cdot Δ t_{c}

velocity after a time(carried, y)

v_{x} (later) = v_{x} (dropped) + a_{x} (f) \cdot Δ t_{f}

velocity after a time(falling, x)

v_{y} (later) = v_{y} (dropped) + a_{y} (f) \cdot Δ t_{f}

velocity after a time(falling, y)

v_{x} (later) = v_{x} (start) + a_{x} (a) \cdot Δ t_{a}

velocity after a time(all along, x)

v_{y} (later) = v_{y} (start) + a_{y} (a) \cdot Δ t_{a}

velocity after a time(all along, y)

x_{x} (dropped) = x_{x} (start) + v_{x} (start) \cdot Δ t_{c} + \frac{a_{x} (c) \cdot {Δ t_{c}}^{2}}{2}

position after a time(carried, x)

x_{y} (dropped) = x_{y} (start) + v_{y} (start) \cdot Δ t_{c} + \frac{a_{y} (c) \cdot {Δ t_{c}}^{2}}{2}

position after a time(carried, y)

x_{x} (later) = x_{x} (dropped) + v_{x} (dropped) \cdot Δ t_{f} + \frac{a_{x} (f) \cdot {Δ t_{f}}^{2}}{2}

position after a time(falling, x)

x_{y} (later) = x_{y} (dropped) + v_{y} (dropped) \cdot Δ t_{f} + \frac{a_{y} (f) \cdot {Δ t_{f}}^{2}}{2}

position after a time(falling, y)

x_{x} (later) = x_{x} (start) + v_{x} (start) \cdot Δ t_{a} + \frac{a_{x} (a) \cdot {Δ t_{a}}^{2}}{2}

position after a time(all along, x)

x_{y} (later) = x_{y} (start) + v_{y} (start) \cdot Δ t_{a} + \frac{a_{y} (a) \cdot {Δ t_{a}}^{2}}{2}

position after a time(all along, y)

{v_{x} (dropped)}^{2} = {v_{x} (start)}^{2} + 2 \cdot a_{x} (c) \cdot (x_{x} (dropped) - x_{x} (start))

velocity after a distance(carried, x)

{v_{y} (dropped)}^{2} = {v_{y} (start)}^{2} + 2 \cdot a_{y} (c) \cdot (x_{y} (dropped) - x_{y} (start))

velocity after a distance(carried, y)

{v_{x} (later)}^{2} = {v_{x} (dropped)}^{2} + 2 \cdot a_{x} (f) \cdot (x_{x} (later) - x_{x} (dropped))

velocity after a distance(falling, x)

{v_{y} (later)}^{2} = {v_{y} (dropped)}^{2} + 2 \cdot a_{y} (f) \cdot (x_{y} (later) - x_{y} (dropped))

velocity after a distance(falling, y)

{v_{x} (later)}^{2} = {v_{x} (start)}^{2} + 2 \cdot a_{x} (a) \cdot (x_{x} (later) - x_{x} (start))

velocity after a distance(all along, x)

{v_{y} (later)}^{2} = {v_{y} (start)}^{2} + 2 \cdot a_{y} (a) \cdot (x_{y} (later) - x_{y} (start))

velocity after a distance(all along, y)

x_{x} (dropped) - x_{x} (start) = \frac{(v_{x} (dropped) + v_{x} (start)) \cdot Δ t_{c}}{2}

mean of the velocities(carried, x)

x_{y} (dropped) - x_{y} (start) = \frac{(v_{y} (dropped) + v_{y} (start)) \cdot Δ t_{c}}{2}

mean of the velocities(carried, y)

x_{x} (later) - x_{x} (dropped) = \frac{(v_{x} (later) + v_{x} (dropped)) \cdot Δ t_{f}}{2}

mean of the velocities(falling, x)

x_{y} (later) - x_{y} (dropped) = \frac{(v_{y} (later) + v_{y} (dropped)) \cdot Δ t_{f}}{2}

mean of the velocities(falling, y)

x_{x} (later) - x_{x} (start) = \frac{(v_{x} (later) + v_{x} (start)) \cdot Δ t_{a}}{2}

mean of the velocities(all along, x)

x_{y} (later) - x_{y} (start) = \frac{(v_{y} (later) + v_{y} (start)) \cdot Δ t_{a}}{2}

mean of the velocities(all along, y)

d (a) = d (c) + d (f)

path in parts(all along)

Δ t_{a} = Δ t_{c} + Δ t_{f}

time in parts(all along)

a (c) = a (a)

steady through its parts(all along, 1)

a (f) = a (a)

steady through its parts(all along, 2)

v_{g,x} (start) = v_{g} (start) \cdot \cos θ_{g} (start)

parts of a velocity(ground, start, x)

v_{g,y} (start) = v_{g} (start) \cdot \sin θ_{g} (start)

parts of a velocity(ground, start, y)

v_{g,x} (dropped) = v_{g} (dropped) \cdot \cos θ_{g} (dropped)

parts of a velocity(ground, dropped, x)

v_{g,y} (dropped) = v_{g} (dropped) \cdot \sin θ_{g} (dropped)

parts of a velocity(ground, dropped, y)

v_{g,x} (later) = v_{g} (later) \cdot \cos θ_{g} (later)

parts of a velocity(ground, later, x)

v_{g,y} (later) = v_{g} (later) \cdot \sin θ_{g} (later)

parts of a velocity(ground, later, y)

v_{x} (start) = v (start) \cdot \cos θ (start)

parts of a velocity(stone, start, x)

v_{y} (start) = v (start) \cdot \sin θ (start)

parts of a velocity(stone, start, y)

v_{x} (dropped) = v (dropped) \cdot \cos θ (dropped)

parts of a velocity(stone, dropped, x)

v_{y} (dropped) = v (dropped) \cdot \sin θ (dropped)

parts of a velocity(stone, dropped, y)

v_{x} (later) = v (later) \cdot \cos θ (later)

parts of a velocity(stone, later, x)

v_{y} (later) = v (later) \cdot \sin θ (later)

parts of a velocity(stone, later, y)

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

magnitude from components(displacement of stone during carried)

x (dropped) = \sqrt{{x_{x} (dropped)}^{2} + {x_{y} (dropped)}^{2}}

magnitude from components(position of stone at dropped)

x (start) = \sqrt{{x_{x} (start)}^{2} + {x_{y} (start)}^{2}}

magnitude from components(position of stone at start)

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

magnitude from components(displacement of stone during falling)

x (later) = \sqrt{{x_{x} (later)}^{2} + {x_{y} (later)}^{2}}

magnitude from components(position of stone at later)

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

magnitude from components(displacement of stone during all along)

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

magnitude from components(average velocity of stone during carried)

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

magnitude from components(average velocity of stone during falling)

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

magnitude from components(average velocity of stone during all along)

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

magnitude from components(change of velocity of stone during carried)

v (dropped) = \sqrt{{v_{x} (dropped)}^{2} + {v_{y} (dropped)}^{2}}

magnitude from components(velocity of stone at dropped from ground)

v (start) = \sqrt{{v_{x} (start)}^{2} + {v_{y} (start)}^{2}}

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

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

magnitude from components(change of velocity of stone during falling)

v (later) = \sqrt{{v_{x} (later)}^{2} + {v_{y} (later)}^{2}}

magnitude from components(velocity of stone at later from ground)

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

magnitude from components(change of velocity of stone during all along)

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

magnitude from components(average acceleration of stone during carried)

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

magnitude from components(average acceleration of stone during falling)

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

magnitude from components(average acceleration of stone during all along)

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

magnitude from components(acceleration of stone during carried)

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

magnitude from components(acceleration of stone during falling)

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

magnitude from components(acceleration of stone during all along)

v_{g} (start) = \sqrt{{v_{g,x} (start)}^{2} + {v_{g,y} (start)}^{2}}

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

v_{g} (dropped) = \sqrt{{v_{g,x} (dropped)}^{2} + {v_{g,y} (dropped)}^{2}}

magnitude from components(velocity of ground at dropped from ground)

v_{g} (later) = \sqrt{{v_{g,x} (later)}^{2} + {v_{g,y} (later)}^{2}}

magnitude from components(velocity of ground at later from ground)

a_{y} (f) = - g

Neglect air resistance