Sidebilder
PDF
ePub
[merged small][merged small][ocr errors][merged small][merged small][merged small]
[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

which we know to be the case from other and more direct principles.

508. Let a and 6 be the semi-axes of the ellipse; then the velocity at the extremity of the minor axis is

b

[ocr errors]

a

P

b
He being the absolute force.
Again, since
Fdę = -

pedo

odo =

[ocr errors][merged small][merged small][merged small][merged small][ocr errors][subsumed][ocr errors]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

2s 2 x du the whole time of descent to the focus.

Again, by 484, the time of the body's moving from the nearer apside to the extremity of the axis minor is T' =

NH

[ocr errors]

Х

(-e).

[blocks in formation]

509. Let R be the distance from the centre of force at which the body is projected; then since odoc

Fdę =

[ocr errors]

V =

[ocr errors]

n

1

Let a = 00, and

§ = R, then fo

1

R~1 But since the velocity in any curve is that which would be ac

)

quired by the bodys's descent along chord of curvature (PV)

[merged small][merged small][merged small][ocr errors][merged small]
[merged small][merged small][merged small][merged small][merged small][ocr errors]

510. If the centre of force be any where about a circle, we learn from (504) that F=

8cre

(ne - a* + 39) r being the radius of the circle, and a the distance of the centre of force from that of the circle.

Now when the force is in the circumference a = r, and p is positive

8cm .::F=

[ocr errors]
[ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Let q = 2r; then

162 2 =

[ocr errors][merged small][ocr errors]

511.

Let the earth’s radius be R, and the space

1

F:9 ::

:

fallen through in a second at its surface; also let p be the periodic time of the Moon. Then if

$

be the distance of the moon from the earth, and F the force of the earth's attraction upon the moon, we have

1

R: or F =

Rpg

ga Now by (440), we get

27

xe

[ocr errors]

Х

[ocr errors]

:: p = R'o pt

and &

R$g} pf

4*** the distance required in feet, g being equal to 32 2

feet, and p expressed in seconds.

512.

2

s= SP;

Generally the time in the parabola is (see 484).

** ( – )* (+ r) go being

= SA and and the time of falling from rest through any space ? - I when FESTA F=is (495).

* {(sv – vyt + (r – vers.- -)} and when x = 0

[ocr errors]
[ocr errors]

2.1

[ocr errors][ocr errors][merged small][merged small]
[blocks in formation]

513. Let x be the angle between the tangent or direction of the body's motion and S; then p = g. sin. y.

dy
= sin. y te cos. it.
de

de

and dp

de : (438).

But us : 0 ::

dp P v and v' being the velocities in a curve, and a circle at the same distance. Hence

1:09 :: sin.y: sin. : sin. y te cos. *.

the

[merged small][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors]

.(a)

Hence, since y cannot exceed 90°, if de be considered positive, dit is positive, or negative, or y is increasing or decreasing, according as v' is > or < than v. If be negative or be decreasing, then dy is positive or negative, that is, 4 is increasing, or decreasing, according as v is > or <v'.

These results indicate a defect in the enunciation of the problem.

[ocr errors]
[ocr errors]

514. Let R be the earth's radius. Then since the velocity of a body revolving at the surface is

✓ R?

R we have for the new trajectory

c= P x V = 6

= x

: odh

a

« ForrigeFortsett »