Dropped From a Truck: Heading

Q. A truck starts from rest and accelerates uniformly at 2.0 m/s². At t = 10 s, a stone is dropped by a person standing on the top of the truck (6 m high from the ground). Neglect air resistance, and take g as 10 m/s². At what angle to the horizontal is the stone moving at t = 11 s, in degrees? expects -26.6°

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

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. At what angle to the horizontal is the stone moving at t = 11 s $Δ t_{a} = 11$ s, in degrees?

The playground

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

direction

θ

The solution

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 $θ (later)$ .

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) $v (later) = \sqrt{{v_{x} (later)}^{2} + {v_{y} (later)}^{2}}$
$v (later) = \sqrt{20^{2} + {(−10)}^{2}}$
$v (later) = 22.3606798$ m/s
parts of a velocity(stone, later, x) $v_{x} (later) = v (later) \cdot \cos θ (later)$
$20 = 22.3606798 \cdot \cos θ (later)$
$θ (later) = -26.57°$
$θ (later) \approx −26.6$ °to 3 significant figures

Conclusion

The stone keeps the truck's velocity sideways while gravity adds one downward, so it heads below the horizontal along the sum of the two.

The Ruby

1require_relative "../physics"23question :dropped_from_a_truck_heading 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}. %{direction} at t = %{asked_at}, in degrees?"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 direction: "At what angle to the horizontal is the stone moving"14end

1require_relative "../questions/dropped_from_a_truck_heading"2require_relative "../science/motion_in_a_plane/motion_in_a_plane"34solve :dropped_from_a_truck_heading, 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  direction = stone.direction(at: later)24  answer direction:, in: "deg", figures: 32526  conclusion "The stone keeps the truck's velocity sideways while gravity adds one downward, so it heads below " \27             "the horizontal along the sum of the two."28end

1require_relative "../solutions/dropped_from_a_truck_heading"23play_with :dropped_from_a_truck_heading do4  title "Dropped From a Truck: Heading"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 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