A particle revolves uniformly in a vertical circle with a velocity (v); find the vertical and horizontal velocity at any point. If the time of one revolution is 8 seconds, and the radius of the circle be 12 inches, and the particle start from the highest point, find the horizontal and vertical velocity after it has revolved for one second.

A body descends uniformly down an inclined plane one mile in length in one hour and twenty minutes; if the plane rises one foot vertical for 100 feet in length, find the vertical velocity of the body in feet per second. 2. How is a uniform force measured? State briefly how it is shewn that the force of gravity at any point on the earth's surface is a uniform force according to the definition.

If a body is projected vertically upwards with a given velocity, find (1) the height that it will ascend in a given time, (2) the greatest height it will ascend.

A body is projected upwards with a certain velocity, and it is found that when in its ascent it is at a point 960 feet from the ground it takes four seconds to return to the same point again; find the velocity of projection, and the whole height ascended.

3. When a heavy particle moves down or up a given smooth inclined plane, find the accelerating or retarding force which acts upon it.

If two vertical circles touch each other at their lowest point, and any straight line be drawn from that point to cut the inner and to meet the outer circle, shew that the time of a heavy particle falling from rest along the part of the line (considered as an inclined plane) intercepted between the circles is constant.

A straight line without a circle and in the same plane with it is parallel to its vertical diameter; find the straight line of quickest descent from the given line to the circle, and determine the angle which the line so drawn makes with the tangent at the lowest point of the circle.

4.

State the second law of motion, and shew how it is applicable to the theory of projectiles.

Shew that the curve described by a projectile in vacuo is a parabola. If (Z) be the latus rectum of the parabola, (R) the range of the projectile, and a the angle of projection, prove R=L tan a.

5. If three bodies are projected simultaneously in the same vertical plane from the same point, prove that the triangle formed by joining the three bodies at any instant of their motion will vary as the square of the time.

If the angles of projection are also the same, shew that at any instant of their motion the bodies will be in a straight line.

6. Give some reasons why the Statical and Dynamical measures of force are differently estimated.

State the law of motion which connects these measures of force, and shew how to obtain one from the other.

A horizontal pressure of 9 pounds acts on a weight (W) along a smooth horizontal table, and after moving through a space of 25 feet from the rest, generates in it a velocity of 10 feet per second; find (W) in pounds.

7. An elastic ball impinges obliquely with a given velocity on a smooth fixed plane, the elasticity between the ball and the plane being given; find the velocity after impact, and the direction of the rebound.

An elastic ball projected at a given angle from a point in a horizontal plane rebounds from the plane; find the range after the first rebound, and the time of flight, the coefficient of elasticity being.

8. When one elastic ball impinges directly on another what kind of mutual action is supposed to take place during the impact, and what ratio resulting therefrom does the coefficient of elasticity express ?

When one elastic ball impinges directly on another compare the relative velocities before and after impact. If the impinging ball be two pounds, and the other ball be one pound in weight, find the coefficient of elasticity when the velocity with which the larger ball impinges is equal to the velocity of the smaller ball after impact.

9. Two inclined planes with a common altitude are placed back to back; (P) and (Q) are two weights, one on each plane, connected by a string which passes over the top of the planes, and P descends drawing up Q; find the accelerating force and the tension of the string.

Phangs vertically, and is 9 lbs.; Qis 6 lbs. on a plane whose inclination is 30°; shew that P will draw Q up the whole length of the plane in half the time that Q hanging vertically would draw Pup the plane.

IO. If a body whose mass is (m) revolve uniformly in a circle, radius

mv2

(r), with a velocity (v), prove that the body is acted on by a force r

tending to the centre of the circle.

A locomotive engine weighing 9 tons passes round a curve 600 feet in radius with a velocity of 30 miles an hour; what force tending towards the centre of the curve must be exerted by the rails so that the engine may move on this curve?

II.

How is the accumulated work or the kinetic energy of a moving body estimated?

A train runs from rest for a mile down a plane whose descent is one foot vertical for 100 feet in length; if the resistances are eight pounds per ton, how far will the train be carried along the horizontal level at the foot of the incline?

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1881.

PRELIMINARY EXAMINATION.

I.

I. EUCLID (Books I.—IV. AND VI.).

Define a plane superficies, a sector of a circle, a rhombus, a rhomboid, a square, a rectangle. What is a rhomboid generally called? Criticise Euclid's definition of a square.

2. If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another.

What other converse proposition may be obtained from Proposition V., Book I.?

3. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, namely, the sides opposite to the equal angles in each, the triangles shall be equal in all respects.

Three given straight lines meet in a given point; shew how a straight line may be drawn to cut them so that its two segments intercepted between the lines may be equal.

4. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of every triangle are together equal to two right angles.

What fractions of a right angle will the angles of a pentagon be, if they are in the ratios of the numbers 1, 3, 6, 9, 11?

5. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the two parts.

How must a straight line be divided into two parts, so that the rectangle contained by them may be the greatest possible?

6. Draw a straight line from a given point, either without or in the circumference, which shall touch a given circle.

Draw the common tangents to two circles which cut one another.

7. Define the segment of a circle,

A segment of a circle being given, describe the circle of which it is the segment.

8. Inscribe a circle in a given triangle.

Inscribe also a second circle in the space intercepted at one of the angles, so as to touch the circumference of the circle and each of the sides containing the angle.

9. Describe an isosceles triangle BAC, having each of the angles at the base double of the third angle BAC.

If a point D be taken in AB so that AD is equal to BC, and DE be drawn parallel to AC to meet BC in E, shew that AB touches the circumscribing circle of the triangle CDE.

10. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

D is any point in AC, the base of an isosceles triangle ABC. DE and DF are straight lines making equal angles with AC, and meeting the equal sides BC and AB in E and F respectively. Prove that the triangles AED, CDF are equal in area.

II. Describe a rectilineal figure which shall be similar to one given rectilineal figure and equal to another given rectilineal figure.

II. ARITHMETIC.

(Including the use of Common Logarithms.)

[N.B.-Great importance will be attached to accuracy in numerical results.]

Б

3. Multiply together 24, 1, 2, and 114 of 15%.

4. Divide 31 of 1 by 51, and the result by 14.

5. Add the difference between 035 of a ton and '064 of a cwt. to the difference between 27 of a qr. and 78 of a lb., and give the answer in lbs. and the decimal of a lb.

6. Multiply 8.07639 by 002873.

7. Divide 298.08 by '00345.

8. Divide 73 by 584.

9. Reduce 03257 of an acre to square yards and the decimal of a square yard.

10. Express 4 ozs. 17 dwts. 12 grs. as the decimal of a lb. troy.

II. Divide 12 miles 2 furlongs 20 poles 4 yards 2 feet 6 inches by 47.

12.

Find the dividend of £3,407. 15s. at 13s. 9d. in the £.

13. At what rate per cent. simple interest will £245 amount to £324. 18s. 73d. in 7 years?

14. A man leaves £32,818 to be divided among his four sons in the proportion of the fractions 2,, and . Find the share of each.

'4'

15. By selling goods for a certain sum a man gains 5 per cent. If he had sold them for 3 shillings more he would have gained 6 per cent. Find their cost price.

16. A buys a pipe of wine and sells it to B at a profit of 5 per cent., B sells it to C at a profit of 5 per cent., and C sells it to D for