Ex. 673. Determine the time of flight and range on a plane inclined at an angle of 10° upward from the horizon in the case of a body projected as in Ex. 668. Ans. 2.88 sec. and 233.6 ft.

Ex. 674. A body is projected with a velocity of 120 ft. per second in a direction making an angle of 28° 45′ with the horizon; determine the time of flight and range on a plane passing through the point of projection--(1) when it is horizontal; (2) when inclined upwards from the horizon at an angle of 12°; (3) when inclined downward at the same angle.

Ans. (1) 3.61 sec. and 379.5 ft. (2) 2.21 sec. and 237.7 ft. (3) 5 sec. and 538.3 ft.

Ex. 675. A body is thrown horizontally with a velocity of 50 ft. per second from the top of a tower 100 ft. high; find after how long it will strike the ground, and at what distance from the foot of the tower.

Ans. (1) 2.5 sec. (2) 125 ft.

Ex. 676. If any number of bodies are thrown horizontally from the top of a tower, they will all strike the ground at the same instant whatever be the velocities of projection.

Ex. 677. There is a hill whose inclination to the horizon is 30°; a projectile is thrown from a point on it at an angle inclined to the horizon at 45°; show that if it were projected down the plane its range would be nearly 3 times what the range would be if it were thrown up the plane.

Ex. 678. In the last Example suppose the slope of the hill to be due north and south, and the azimuth of the plane of projection to be a; show that the sum of the two ranges obtained by throwing the body towards the ascending and descending parts of the hill equals

[The azimuth is the bearing of a point from the south measured on a horizontal plane ]

Ex. 679. If there are two inclined planes and the angle between them is bisected by the horizontal plane, and if the ranges of the same projectile on the three planes are R1, R2, and R respectively, show that

R1 + R2: R::2: cos inclination

Ex. 680. If in Example 672 the body is so projected as to obtain the greatest range with a given velocity, show that the direction of projection must bisect the angle between the vertical and the plane.

[It must be remembered that 2 sin (a-0) cos a=sin (2a-0) - sin 0.] Ex. 681.-Referring to the figure in Prop. 25, if an and HP are denoted by x and y respectively, show that

▼2

2g

x=vt cos a

y=vt sin a-gt2

Ex. 682. Show that the highest point the projectile can reach is

sin a feet above the middle point of the horizontal range.

Ex. 683.-Two bodies of unequal weight are thrown from the same point in different directions with different velocities; find the position of their centre of gravity after t seconds.

Proposition 26.

The curve described by a projectile in vacuo is a parabola whose directrix is horizontal, and at a height above the point of projection equal to that to which the velocity of projection is due.

FIG. 161.

Let P be the point, and PM the direction of projection; let PQ be the path of the projectile, and q its position at the end of t seconds; draw the vertical lines DPN and MQ, also draw QN parallel to PM, then

D

I

N

M

Q

PN=QM=gt2
N=PM=vt

2v2
... QN2= PN
g

Now, this relation between QN and PN is the same, wherever on the curve we may take q; but if a parabola were drawn through P touching PM, with its diameter vertical and its directrix passing through a point D so taken that 4 PD

i. e. it would coincide with the curve described by the projectile: hence that curve is a parabola whose directrix is horizontal and passes through the point D; but it will be remarked that

v2=2g.DP

So that DP is the height to which the velocity of projection is due. (See Art. 116.)

Cor. The velocity of the projectile at any point is that due to the height of the directrix above that point.

Let PQP' be the path of the projectile, of which DD' IS the directrix; at the point q let the body be moving with a velocity v in the direction QT; now, it is plain that if another body were thrown from q in the direction or with an equal velocity v, it would move in exactly the same manner as the projectile, i. e. it would describe the curve

FIG. 162.

D

C

Q

D

T

QP'; but if a body is thrown from q so as to describe that

P

P

curve, it must be thrown with the velocity due to the height qc, i. e. v2=2gcq.

Ex. 684. In fig. 159, show that if BN is divided into any number of equal parts, and the points of section joined to a, the curve will be divided into parts that are described in equal times.

Ex. 685. Show that the velocity v of the projectile at any time t is given by the formula

v2=v2 - 2v gt sin a+g2t2.

Ex. 686. There is a wall b feet high, a body is thrown from a point a feet on one side of it so as just to clear the wall and to fall c feet on the other side: show that

128. The First and Second Laws of Motion. The object

of the first law of motion is to assert that a body has no

6

power of changing its own state of rest or motion, and that every such change is due to the action of some external force. Up to the time of Galileo it was supposed that certain kinds of motion such as the rolling of a body along a road-have a natural tendency to decay; while certain other kinds such as that of falling bodies have a natural tendency to increase. When this opinion came to be examined, it was found that every case of 'decay' could be referred to the action of retarding forces, e. g. friction and resistance of the air, and that the 'decay' could be made indefinitely slower by diminishing these resistances; on the other hand, every case of increased velocity could be referred to the action of an accelerating force such as gravity. The law is stated as follows: 'A body not acted on by any external force, if at rest, will continue at rest, and if in motion will continue to move uniformly in a straight line.' The object of the second law of motion is to assert that the effect produced by a force is irrespective of the previous motion of the body; it is enunciated thus: 'When a force acts on a body in motion, the velocity it would produce in the body moving from rest is compounded with the previous velocity of the body.' If the body is moving along the line of action of the force, the term compounded must be understood to mean added (or subtracted); if the body is moving transversely to the line of action of the force, the word compounded must be understood as in Art. 127. The principle asserted in the second law of motion is illustrated by many well-known facts, such as the following: A person on board a ship can throw up a ball and catch it with equal facility whether the ship is at rest or in a state of steady motion.

CHAPTER III.

ON FORCE AND MOTION.

129. Acceleration produced by a given Force. The following Examples can be solved by means of the principle laid down in Art. 118.

Ex. 687. If a body slides down a smooth inclined plane, show that the acceleration equals g sin a, where a is the inclination of the plane to the horizon; if the plane is rough, show that the acceleration equals

g

sin (a-4) where is the limiting angle of resistance.

cos φ

[In the case of a smooth plane the force producing motion is the part of its weight resolved along the plane, i.e. w sin a, whence f=g sin a. In the case of the rough plane the force producing motion is w sin a sin (a-4) whence

diminished by the friction, i.e. w sin a-uw cos a or w

cos φ

Ex. 688. Find the velocity acquired by a body in descending a smooth inclined plane 50 feet long and having an inclination of 23°; determine also the velocity that would be acquired if the limiting angle of resistance were 15°. Ans. (1) 35.4 ft. per sec. (2) 21.5 ft. per sec.

Ex. 689. If a body begins to ascend an incline, show that the retardation is g

sin (a + $)
cos φ

Ex. 690. There is a plane 50 ft. long and inclined to the horizon at an angle of 30°; the limiting angle of resistance between it and a given body is 15°; determine the velocity the body must have at the foot of the plane so as just to reach the top, and the time it will take to get there.

Ans. (1) 48.4 ft. per sec. (2) 2.07 sec.

Ex. 691. A body just rests on a plane which is inclined at an angle of 30° to the horizon. Find the velocity acquired and the space described from rest by the body when the plane is inclined at an angle of 60° to the horizon. Ans. (1) 36.9 ft. per sec. (2) 36.9 ft.