IX. DYNAMICS.

I. How is velocity measured, (1) uniform, (2) variable?

I

The Derby course is 1 mile 5 furlongs in length. If the race is run in 2 minutes 30 secs., compare the average velocity of the winning horse with that of a train moving at the rate of 40 miles an hour.

2. A particle starts with a velocity of 20 yards a minute, under a uniform acceleration, producing an increase of velocity of 100 miles an hour in each hour, find the number of feet described by it in 6 seconds.

3. A projectile is projected with velocity V from a point in a plane which is inclined at the angle a to the horizon in a direction making an angle ẞ with the plane measured upwards: find the range on the plane.

If the plane be smooth, find the greatest distance measured along the plane to which the particle will ascend, and the time before it again reaches the point of projection.

If a given impulse acting on a stone causes it to rise to a height h, how high will a stone of half the weight of the former rise under the action of the same impulse?

5. Assuming that the resistance of all kinds to a train on the level is equal to 6 lbs. per ton of the total weight of the train, when it is moving with less speed than 10 miles an hour, with an addition of lb. per ton for every mile of speed above 10, that the total weight of a train and engine is 200 tons, and the maximum speed obtainable 40 miles an hour, find the time from rest of reaching a velocity of 10 miles an hour, the engine always exerting its full force.

6. A person wishes to throw a stone so as to produce the greatest possible blow, at a point in a smooth vertical wall, at a height / from the ground. His strength is sufficient to throw the stone vertically upwards to a height 2h. Prove that he must throw from a point distant 2h from the foot of the wall, the resistance of the air and height of his hand in throwing being neglected.

7. Explain generally the reasoning by which it is proved that the time of oscillation of a simple pendulum of length / is π

If a seconds pendulum be formed of a heavy particle suspended by a string of length (1) from a point A, in a vertical wall, and if a nail jut out from the wall at a distance vertically below 4, so as to catch the string

n

once in each complete oscillation, find the time of a complete oscillation (n being large).

8. Find the acceleration of a body on a smooth inclined plane.

Prove that the time of falling down the plane from rest is the same as the time of moving over the same distance, with a uniform velocity equal to half that acquired in falling down the plane.

9. Define angular velocity, and state how it is measured. Find the linear velocity and acceleration of a point on the circumference of a circle of radius (a) revolving with angular velocity (w) round the centre fixed.

If the centre be at the same time moving along a given straight line with uniform velocity (μ) find the velocity and acceleration of any given point on the circumference.

10. Find, after direct impact, the velocities of two spheres of equal radii but different masses, their elasticity being given,

If the impact be oblique and if be the inclination of the relative velocity to the line of centres before impact: find the corresponding inclination after impact.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

JULY, 1885.

PRELIMINARY EXAMINATION.

I. EUCLID (Books I.-IV. AND VI.).

[Ordinary abbreviations may be employed; but the method of proof must be geometrical. Great importance will be attached to accuracy.]

I. 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, viz., sides which are opposite to equal angles in each; then shall the other sides be equal, each to each, and also the third angle of the one to the third angle of the other.

ABCD is a rectangle, and AE, BF are drawn to meet the diagonals BD, AC in E and F respectively, and in such a direction that the angles AEB, AFB are equal to one another. Show that the triangles AEB, AFB are equal in all respects.

2. The opposite sides and angles of a parallelogram are equal to one another, and the diameter bisects the parallelogram, that is, divides it into two equal parts.

AB, CD, EF are three parallel straight lines: and the points A, C, E are in a straight line, and also the points B, D, F. Prove that if AC is equal to CE, then BD is equal to DF.

W. P.

I

I2

3. If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced.

4. In obtuse angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by the side on which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle.

What is the value in similar terms of the square on the side subtending one of the acute angles of this triangle?

ABCDE is a straight line so divided that AB=BC=CD=DE, and O is an external point; show that the difference of the squares on OA and OE is twice the difference of the squares on OB and OD.

5. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

If any eight-sided figure be inscribed in a circle, show that the sum of either four alternate angles is equal to six right angles.

6. If from a point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square on the line which touches it.

Two equal circles intersect in A and B; show that if from an external point a tangent be drawn to each circle, the tangents will be unequal unless O lies on AB produced.

7. Inscribe a circle in a given triangle.

Find that straight line which would, if produced, bisect the angle between two given straight lines, without producing the given straight lines to meet.

8.

Describe an isosceles triangle having either of the angles at the base double of the third angle.

9. What is Euclid's test for Proportion?

Triangles and parallelograms of the same altitude are to one another as their bases.

IO.

In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.

If AD be the perpendicular, and AB, AC the two sides including the right angle: prove that

II. Similar triangles are to one another in the duplicate ratio of their homologous sides.

II. ARITHMETIC.

(Including the use of Common Logarithms.)

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

3.

33 02083 11'89

A cube has an edge 2 ft. 6 in. long: find the ratio between the sum of the areas of the semicircles described on its edges, and the whole surface of the cube, having given that the area of a circle = 3*1416 times the square of the radius.

4. Extract the square root of 02010724, and the cube root of 18.609625.

5. Find the value of 54 of 8s. 3d. + 027 of £2. 155. +3125 of £2. 25.

6. If 24 oxen require 6 acres of turnips to supply them for 10 weeks, how many acres would supply 6 score of sheep for 15 weeks, 3 oxen eating as much as 10 sheep?

7.

What weight must be added to of of half a cwt. to make it up to of 3 quarters avoirdupois ?

8. Reduce 155. 9åd. to the decimal of £1, and £5. 7s. 6 d. to the decimal of one shilling.