Spencer Heath Archive

Item 54

Penciled by Heath on lined notepad paper

No date

GRAM-CENTIMETER-SECOND ANALYSIS

When a gram accelerates from 0 to 1 centimeter per second in one second it takes on an average velocity of ½ centimeter per second during that second. The force required for this is called a dyne. The energy taken on, then is 1 dyne times ½ centimeter or ½ erg — since erg is the name for a dyne-centimeter.

This force of one dyne continuing for another second will cause the gram to move another ½ centimeter during the second second. But during this second second the gram will move also a whole centimeter due to the velocity acquired during the first second. The gram will therefore move during the first two seconds two half centimeters due to two accelerations and one full centimeter due to the velocity acquired during the first second /?/.

In the first second the gram will move ½ centimeter and have at the end of that second a velocity of one centimeter per second.

During the second second the velocity will move it one centimeter and a second acceleration will cause it to move a second ½ centimeter in addition to the ½ centimeter that it moved during the first acceleration.

These two half centimeters taken together with the whole centimeter due to velocity during the second second amounts to a total movement of two centimeters during the first two seconds. The energy acquired by the gram will therefore be two dyne-centimeters or two ergs.

Velocity = Centimeters

                Seconds

Let X = total centimeters

First second would be ½ centimeter

Last second would be ½ centimeter

In any second it moves a distance of one centimeter for each velocity already acquired and ½ centimeter in acquiring a new velocity and it has moved —

1. The first ½ centimeter
2. A ½ centimeter for each previous second
3. A centimeter

Based on its present velocity, how far has it moved?

Say V = X centimeters per second. X-l = number of centimeters moved by reason of previously acquired velocities.

½ = number of centimeters gained in first second.

½ = number of centimeters gained in last second.

During each second it keeps the velocity it has acquired and picks up a new one.

Each pick-up is a half centimeter

Let X = number of seconds

X x 1 = distance due to velocity

½ = distance due to acceleration

X x 1 + ½ = centimeters in Xth second

(X-l) x 1 = previous distance

X + ½ + X-1 = 2X – ½

 

Dyne-centimeter is an ERG. But a dyne only moves a gram one half centimeter in a second. So, raising the velocity of the gram to one centimeter per second in one second requires only one-half erg because the distance is only one-half centimeter.

During the next second the distance due to velocity is two one-half centimeters and to acceleration one half centimeter, as in the first second. So the total distance in 2 seconds is 2 centimeters — Energy 2 dyne-centimeters or 2 ergs.

If the motion the first second had been a whole centimeter (the dyne being strong enough) then the distance moved in 2 seconds would have been 4 cms instead of 2. Then the square of the number of seconds would have been 4 and the dyne-cms also 4. The ergs would have agreed always with the square of the number of seconds.

But because the dyne only moves the gram half a centimeter the energy has to be taken as half the square of the velocity.

 

Seconds	Distance	Velocity
1st	½	1
2nd	2	2
3rd	4½	3
4th	8	4
5th	12½	5

 

Since the velocity is always increased one centimeter per second for each second, the number of seconds is the same as the velocity.

(Using motion as 1 cm the first second)

Each unit increment of time makes the square of the time larger than it was by a number of units always 2 more than the preceding unit or addition of units.

The amount of motion or distance added always corresponds

/Photocopy page here/

(Assuming one whole unit of distance in 1st second)

Units of Distance

 

1st Second 1 unit only

2nd Second Carried forward                 ½   1 unit

 

Gained by velocity                1   2

Added by acceleration               ½   1

                                               —

3rd          Carried forward              2   4

            Gained by velocity (now doubled)2   4

    By acceleration               ½   1

                                           —

4th              Forward                4½   9

By velocity (trebled)             3   6

By acceleration                  ½    1