ACCELERATE FOR A DAY
Albert Einstein loved thought experiments.
EQUIVALENCE: Perhaps one of his more famous ones was about an elevator which accelerates at same rate as a free falling object near Earth's surface (This rate is well known as g, 9.80665 meters per second per second (m/sec^{2}); for convenience, we round this value to 10 m/sec^{2}.)
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 "gforce" acceleration. 
Define acceleration as velocity per time:
a = v/t
Determine velocity with well known equation:
Let a = g (= 10m/sec^{2});
then, readily compute values (see Table).
Constant gforce
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, we can 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/sec^{2} = g
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 freefall velocity varies with time, not with mass. In fact, numerous observations have determined that an object's speed increases (accelerates) as the freefall time increases. Precise observations have determined this acceleration rate to be 9.80665 meters per second per second (m/sec^{2}); for convenience, our thought experiment will round this value to 10 m/sec^{2}. Thus, we now know that near Earth's surface, an object increases its downward speed an additional 10 m/s for every second of freefall. After 2 seconds of freefall, 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 a counterforce which impedes 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 gforce powered spaceflight. 

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 home. Isaac Newton was born the same year. 
Kilometers per second per day
After 86,400 seconds (one day) of gforce acceleration:
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} = g * t
V_{fin} = 864 km/sec /day * 2 days
V_{fin} = 1,728 km/sec
Table shows other values.
Why express g as "km/sec /day"? Quickly determine inflight velocities as kilometers per second (kps), 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 gforce acceleration, spacecraft attains tremendous speed many times this.
 
Astronomical Units/day per day
For an even more convenient unit of g, consider 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, thought experiment rounds AU to 150,000,000 km.
Different g expression enables different velocity expressions.
After one day of gforce acceleration:
Interplanetary gforce flights will be quick, a matter of days. Therefore, g = 0.5 AU/day^{2} might be a convenient expression. 
Galileo showed that the freefall 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. If the acceleration is now a (less than near Earth gravitational acceleration, g=10m/sec/sec) and the time, t, was measured in clicks. Starting with the ball at rest (D_{0} = 0), one measures following:
For more on this experiment, see the book "The God Particle"
by Nobel prize winner, Leon Lederman, with Dick Teresi.  
Percent Light Speed per day
Finally, consider g in terms of %c, percent light speed.
First, convert c from the well known kilometers per second to AUs per day.
Recall that gforce 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.


Above expressions have nearly equal values; all approximate g, acceleration due to Earth surface gravity. Recall that g is an acceleration which is velocity per time. Above expressions are all stated as velocity per unit time.
0 Comments:
Post a Comment
Links to this post:
Create a Link
<< Home