PROJECTILES IN A VACUUM.

Hence, making the altitude of the tower a, since by the

373.

v2

The space due to a velocity of 744 feet in a second is

(144)

S

644: 58 feet nearly.

Also the space described in two seconds by a body projected upwards with that velocity is

374.

At the end of the time (t) the positions of the bodies

Hence, if (y", x") denote the co-ordinates of the centre of gravity, since

whose greatest ordinate, corresponding abscissa, and paramete (found by putting dy" — 0, &c.) are respectively

(bB + aA)2 sin. 0)

(bB + aA)2 sin. 20

2g (A + B)2

2g (A + B)2

375.

Let a be the altitude of the tower. Then (370)

X

v sin. 20√ v* sin.o 20 + 4gav2 (1 + cos. 28)

2g

..v sin. 20 + √v2 sin.o 20 + 4ga (1 + cos. 20) = max. by the question; which gives by the rule, after proper reductions,

determining the direction required.

Let the point of projection be situated in the horizon; then a=0

376.

Making the diameter (2r) of the circle horizontal,'

we have (370) and by the question,

or the velocity of projection is that acquired down — r.

2

Also (by 370) the greatest ordinate corresponding abscissa, and parameter of the parabola, are respectively

377.

The greatest ordinates and corresponding abscissæ of the parabolas described are (370,) if 20 and 0 are the angles of

Hence (370) the equation to the tragectory is

which is a parabola whose greatest ordinate, corresponding abscissa and parameter are (370)

Let r be the radius of the O; then the velocity (v)

380. acquired down it is

v = √2gr

and the time

Now being the inclination of the plane to the vertical, we have

according as the body is projected downwards or upwards.

..s = 2r ±r cos.

=r+r (1± cos. 0).

Let the negative sign be taken, then

srr vers. 0

Q. E. D.

381. Lev, v be the required velocities of projection: then, generally, s, s', being the spaces described in the time t along the planes whose inclinations to the horizon are 0, 0, with these velocities we have

according as the bodies are projected downwards or upwards.

v

From this result it appears that the ratio may be obtained,

made a corresponding oversight in the enunciation. When =

=

√2