A Thought Experiment: ACCELERATE FOR A DAY

CONTENT
FINAL VELOCITY
AVERAGE VELOCITY
DISTANCE TRAVELED
DAILY RATE: KPS per DAY
DAILY RATE: AU/DAY per DAY
DAILY RATE: %c per DAY
RELATIVISTIC EFFECTS
SIDEBARS
"FORCE AT A DISTANCE"
GALILEO'S MOST FAMOUS EXPERIMENT
GALILEO GALLILEI
GALILEO'S MOST FAMOUS INVENTION
GALILEO FACES THE INQUISITION
GALILEO'S FALLING MOTION EXPERIMENT

Albert Einstein loved thought experiments.

Perhaps the best known thought experiment uses an elevator which accelerates at same rate as a free falling object near Earth's surface. The occupants can't ascertain whether the elevator is accelerating in space or remains static on Earth's surface; thus, Einstein's famous EQUIVALENCE PRINCIPLE.

"FORCE AT A DISTANCE"

In the late 1600's, Newton realized that objects remain at constant velocity until a force acts on them to change their velocity. He further discovered that gravity (attraction between masses) acts as a force at a distance and causes objects to accelerate.

Much later, Albert Einstein gave considerable thought to the accelerating elevator concept and realized that occupants inside the elevator would not discern whether that elevator was static on Earth or accelerating at g in space. Either way, they'd be pressed against the floor by same force which gives us our weight on Earth's surface. For example, a dropped ball would fall at same rate with same trajectory. Thus, Thought Experiments quickly led Einstein to conclude that gravity and acceleration are closely related. (Note: Acceleration is one of the basic units of motion.)

Perhaps we can slightly modify Einstein's experiment and replace the elevator with a notional spaceship which can freely accelerate at rate g throughout the Solar System. Because the spaceship occupants will experience simulated gravity throughout this flight, we'll take the liberty of calling this "g-force" acceleration.

FINAL VELOCITY

V_Fin ≈ t ×	10 m sec²

Elapsed
Time (t)

Final
Vel. (V_Fin)

seconds

meters/sec

......

100

1,000 (=1 km)

1,000

10,000(=10 km)

Given

g × t

Define acceleration as velocity per time:

a = v/t

Determine velocity with well known equation:

v = a × t
thus, velocity equals
acceleration times duration.

Let a = g (≈ 10m/sec²);
then, readily compute values (see Table).

Constant g-force
increases velocity at a linear rate;
achieve final velocity (V_Fin)
after elapsed time of t seconds.

AVERAGE VELOCITY

To determine the average speed of a constantly accelerating object, use half of the final velocity if and only if our acceleration complies with the following conditions:
1. Start at rest, or initial velocity equals 0

v_Ini = 0 m/s

2. Acceleration is constant, or velocity increases at steady rate

a = 10 m/sec² = g

V_Ave =V_Fin / 2
Elapsed Time (t)	Final Vel. (V_Fin)	Ave. Vel. (VAve)
seconds	km/sec	km/sec
1	.01	.005
10	0.1	.05
100	1.0	0.5
1,000	10	5.0
3,600 (1 hr)	36	18
86,400 (1 dy)	864	432
Given	g × t	V_Fin/2

Thus, we achieve average velocity exactly halfway through the acceleration interval; so, compute average velocity by dividing final speed by two.

V_Ave =	V_Fin 2

Given constant acceleration and initial velocity of zero,

average velocity for any duration is half of final velocity.

GALILEO'S MOST FAMOUS EXPERIMENT

... at the Leaning Tower of Pisa showed that heavier objects fall at the same rate as lighter objects.

In fact, he did numerous experiments in a more practical fashion, rolling balls down sloping troughs at different angles. He discovered that an object's free-fall velocity varies with time, not with mass. In fact, numerous observations have determined that an object's speed increases (accelerates) as the free-fall time increases.
Precise observations have determined this acceleration rate to be 9.80665 meters per second per second (m/sec²); for convenience, our thought experiment will round this value to 10 m/sec².

Thus, we now know that near Earth's surface, an object increases its downward speed an additional 10 m/s for every second of free-fall. After 2 seconds of free-fall, object's speed is 20 m/s; after 3 seconds, 30 m/s; and so on. If the shape of an object interferes with the fall (like a feather or a parachute), then air resistance acts as a counterforce to impede the gravitational force for much slower speed.

SUMMARY: Friction free objects with different masses (such as a small ball bearing and a much larger bowling ball) increase velocity a the rate of g, 10 meters per sec for every second of free fall, or spacecraft increases velocity for every second of g-force powered spaceflight.

GALILEO GALLILEI

...was born in Pisa, Italy on February 15, 1564. His family were of nobility but not rich.

In 1581, Galileo began studies at the University of Pisa, where he learned Aristotelian physics which held that heavier objects fall faster through a medium than lighter ones. Galileo eventually disproved this idea and asserted that all objects, regardless of density, fall at the same rate in a vacuum.
To determine this rate, Galileo performed various experiments with dropped objects from a certain height. In an early experiment, he rolled balls down a gently inclined plane; then, he determined their positions after equal time intervals. He wrote down his discoveries about motion in his book, De Motu, "On Motion."

DISTANCE TRAVELED

d = t × V_Ave
Elapsed Time	Final Speed	Ave. Speed	Total Dist.
t	V_Fin	V_Ave	d
1 sec	10 m/s	5 m/s	5 m
2 sec	20 m/s	10 m/s	20 m
3 sec	30 m/s	15 m/s	45 m
. . .	...	...	...
10 sec	100 m/s	50 m/s	500 m
Given	g × t	V_fin/2	t × V_ave
	g×t	g×t/2	g×t²/2

Gravity accelerates freely falling objects 10 meters per second for every second of flight. After 3 sec, object's speed is 30 m/sec. However, the distance fallen will reflect object's average speed. For example, it falls 5 meters in first second of free fall due to its average speed of 5 meters per second.

After three seconds, the object achieves an average speed (V_Ave) of 15 meters per second; multiplying this value by 3 seconds duration gives us 45 meters distance.

ALTERNATE DISTANCE METHOD

Since 10m=.01km, let
g ≈ .01 km/sec²

t t² d

Time
(sec) Distance
(km)

10

100

.5

60

3,600

18

3,600

1.296×10⁶

64,800

86,400

7.464×10⁹

37,324,800

t

t²

g × t² /2

Determine distance traveled with another simple equation.
Recall: V_Ave= V_Fin/2
Thus, d= V_Ave× t = t(V_Fin/2).

Recall: V_Fin = a × t

d = t(V_Fin/2) = t (a × t/2)
d = (a × t²)/2.

Use a = g = .01 km/sec²;

then, g-force distances can be determined:

d = g × t²/2

Final data row uses elapsed time for an entire day (86,400 sec); thus, g-force distance is for an entire day. (Recall that one Astronomical Unit (AU), average distance from Sol to Terra, is about 150 million km. Thus, traveling at g-force for one day takes us about 1/4 AU. See below for more details.)

GALILEO'S MOST FAMOUS INVENTION

... was the telescope. In 1609, Galileo’s first telescope was modeled after previous telescopes that could magnify images three times. He soon created another telescope which could magnify twenty times.

With this improved telescope, he could observe features on the moon, observe a supernova, verify the phases of Venus, discover sunspots and discover the four largest satellites of Jupiter.

His discoveries confirmed the Copernican system; Earth and other planets revolve around the Sun. This starkly contrasted with the view of the Church which then held the Universe to be geocentric.

Galileo's belief in the Copernican System eventually got him into trouble with the Catholic Church’s Inquisition which was charged with the eradication of heresies. A Church committee of consultants declared this Copernican proposition to be heretical. Because Galileo supported the Copernican system, he was warned to not discuss or defend Copernican theories.

GALILEO FACES THE INQUISITION

In 1624, Galileo understood from Pope Urban VIII that he could write about Copernican theory as long as he treated it as a mathematical proposition; thus, he felt safe to publish Dialogue Concerning the Two Chief World Systems. However, certain mean spirited enemies of Galileo persuaded the Pope to interpret certain passages as personal insults. Thus, Galileo was called to Rome in 1633 to face the Inquisition. Galileo was found guilty of heresy for his Dialogue …, and was sent to his home near Florence to be under house arrest for the rest of his life. Though disappointed by the verdict, this distraction free environment enabled Galileo to produce yet more scientific work. In 1642, Galileo died at his home. Isaac Newton was born the same year.

kilometers per second (kps) per day

**Express g differently**
g	=	10 m sec²	×	1 km 1000 m
g	=	.01 kps sec	×	86,400 sec day
g	=	864 kps day

After 86,400 seconds (one day) of g-force acceleration, final velocity is 864 km per second:

V_fin = g × t = 864 km/sec

For every day of constant acceleration, g; spacecraft increases its velocity another 864 km/sec. Thus, we can restate g as a daily rate.

g = 864 km per sec / day

Thus, at end of 2nd day of spaceflight,

V_Fin = t × 864 kps/day
Elapsed Time (t)	Final Vel. (V_Fin)
days	kps
1	864
2	1,728
3	2,592
..	......
10	8,640
Given	g × t

V_fin = g × t

V_fin = 864 km/sec /day × 2 days

V_fin = 1,728 km/sec

Table shows additional values.

Why express g as "kps/day"? Quickly determine inflight velocities as kilometers per second (kps), which is typically used for orbital velocities of Solar System objects (planets, asteroids, comets). For example, Earth's orbital velocity around Sol is about 30 kps. NOTE: After just one day of g-force acceleration, spacecraft attains tremendous speed many times this.

Astronomical Units (AUs)/day per day

Thought Experiment (TE) considers the Astronomical Unit (AU), average distance from Earth to Sol. Distance to nearby planets is conveniently measured in AUs.

Exact AU = 149,597,870 km; for convenience, TE rounds AU to 150,000,000 km.

**Express g differently**
g	=	864 kps day	×	86,400 sec day
g	=	74,649,600 km day²	×	1 AU 149,597,870 km
g	≈	.5 AU day²

Different g expression

enables different velocity expressions.

After one day of g-force acceleration:

V_Fin = t × g V_Fin = 1 day × .5 AU/day² V_Fin = .5 AU/day V_Ave = V_Fin /2 V_Ave = .5 AU/day ÷ 2 V_Ave = .25 AU/day
Elapsed Time (t)	Final Vel. (V_Fin)	Ave Vel. (V_Ave)	Ttl Dist. (d)
day	AU/day	AU/day	AU
1	.5	.25	.25
2	1.0	.5	1.0
3	1.5	.75	2.25
. . .	...	...	...
10	5.0	2.5	25
Given	g × t	V_Fin/2	t × V_Ave
	g×t	g×t/2	g×t²/2

Interplanetary g-force flights will be quick,
a matter of days.
Therefore,
g ≈ 0.5 AU/day²
might be a
convenient expression.

GALILEO'S FALLING MOTION EXPERIMENT

Galileo showed that the free-fall motion of an object has a constant acceleration. Starting from rest, distances increase in proportion to the square of the elapsed time.

Since a vertical fall was too fast for Galileo to measure accurately, he slowed it down by making the ball roll down an inclined board. Across the board, along to its surface, he strung a number of taut horizontal wires, making the ball sound a click whenever it jumped over one of them. Galileo then moved the wires up and down the board, until the clicks sounded evenly spaced.
The board's slope slows ball's velocity to much less than free fall, but Earth's gravity still pulling it downward. The time, t, is measured by the audible clicks. Starting with the ball at rest (D₀ = 0), one measures following:

After one click,	D₁= 1	unit
After two clicks,	D₂= 4	units
After three clicks,	D₃= 9	units
After x clicks,	D_x= x²	units

Galileo confirmed the ratio between the distances to be that of the squares: 1, 4, 9, 16, 25… If one converts clicks into seconds and accounts for the angle of the board as well as for the rolling friction of the ball (not available in Galileo's day), one could even calculate Earth's gravity, g, from observed acceleration, a.

For more on this experiment, see the book "The God Particle"
by Nobel prize winner, Leon Lederman, with Dick Teresi.

Percent Light Speed (%c) per day

Finally, consider g in terms of %c, percent light speed.

**Express light speed, c, differently**
c	=	299,792.458 km sec	×	86,400 sec day
c	=	299,792.5×86,400km day	×	1 AU 149,597,870km
c	≈	173.145 AU day

First, convert c from the well known kilometers per second to AUs per day.

**Express g in terms of c**
g	≈	10 m sec × sec	=	.5 AU day × day
g	≈	.5 AU/day day	×	c 173.145 AU/day
g	≈	.00289 c day	=	.289% c day

Recall that g-force acceleration equals 0.5 AU/day; thus, we can readily transform this value into a percentage of light speed.

Recall that %c is a velocity, and g is an acceleration (velocity per unit time); thus, another value for g is 0.289%c/day.

Do we care about our spacecraft's light speed??

Relativistic effects might be a concern for very fast inflight speeds.

RELATIVISTIC EFFECTS

V_Fin = t × .289%c/day
Elapsed Time (t)	Final Vel. (V_Fin)
day	%c
1	0.289
2	0.578
3	0.867
..	......
10	2.890
Given	g × t

1. For interplanetary flights, we don't care about relativistic effects.

Even at g-force acceleration, ship's speed never exceeds 10% c throughout the Solar System.

Such flights will take a few days, and table shows that we don't even reach 1% c until after three days of g-force flight. NOTE: Due to relativistic effects, our g-force spaceship will never ever reach light speed, c; even though, it can get ever closer and closer. Thus, short time measures of g-force accelerated velocities can be approximated by adding (as shown in table). However, accuracy requires different means of calculation for longer duration flights.

2. On the other hand, relativity will definitely affect interstellar flights. Even with g-force acceleration, those flights will take years. After 300 days of g-force acceleration, we will have not even approached the Oort Cloud, but we speculate that spaceship's velocity will be .866c. At this relativistic speed, spaceship's mass doubles, and inflight time shrinks by 50%. By the time, we'll doing interstellar missions, we'll likely have some solutions for this phenomena.

SUMMARY: Equivalent Expressions of g