The following table gives the speed of a train seconds after leaving a station for portion of the journey between two stations and , ft. apart.
|Speed in feet per second.|
|Time in seconds.|
Estimate the acceleration after seconds and the distance travelled in the seconds. If the train continues to move with the uniform speed of ft. per second until the brakes are applied, producing a uniform retardation of ft. per sec. per sec., find the total time taken in going from to .
(a) A particle describes simple harmonic motion of amplitude , the period being . Prove that at distance from its mean position its speed is given by .
(b) If the amplitude of the motion of bob of a simple pendulum is foot and the period seconds, find the time occupied by the bob in passing between points which are distance ” and ” from the mean position and are on the same side of it.
Prove that two bodies moving in different straight lines with uniform velocities will meet if their relative velocity is in the straight line joining them.
A ship steaming in a direction North of East is seen by a submarine which is in a direction East of South from the ship. Find in what direction a torpedo travelling at miles per hour must be fired in order to strike ship.
A bullet of mass oz. travelling horizontally with a speed of ft. per second becomes embedded in a block of wood of mass lb. suspended by a light steel wire, the C.G. of the block being ft. below the point of suspension. Find the vertical distance through which the block rises.
When the motors of an electric train of effective weight tons are working at the rate of H.P., a uniform speed of miles per hour is attained on the level. Assuming the frictional resistances to remain the same, what is the maximum speed attainable with H.P. when ascending an incline of in ?
Show that the centre of gravity of equal particles placed at the three vertices of a triangle respectively coincide with the centre of gravity of the area of the triangle. The parallel side of a trapezium are of lengths and and the perpendicular distance between them is . Find the distance of the C.G. from the side .
What are the laws of solid friction?
A body of weight is dragged along a horizontal plane, coefficient of friction , by a cord inclined at an angle to the horizontal; find the tension in the cord. For what value of is the tension a minimum? A body is placed on a plane, coefficient of friction inclined at to the horizontal; find the velocity acquired in one second.
A triangular roof-truss has a horizontal span of feet and angle is , and being of equal length. The roof-truss is hinged at and simply supported on rollers at (so that the supporting force at is vertical). It is loaded as follows: tons perpendicular to at its mid-point at . Find the reactions of the supporting forces at and and the tensions in .
A body of weight lb. is moving due East with a velocity of ft. per second, and forty seconds later it is moving North-East with a velocity of ft. per second. What is the change in velocity? What constant force acting during the interval would produce the change? Show that in the latter case the hodograph is a straight line.
Assuming that the wheels of a motor-car are feet in diameter, what is the angular velocity of a wheel when the car s travelling at the rate of miles per hour? What is the velocity of a point on the surface of tyre (a) when vertical above the centre, (b) when the radius from the point to the centre is inclined at an angle of to the upward vertical? What would be the force acting on a stud of mass oz. embedded in the type when at its greatest height?
State Examinations Commission (2018). State Examination Commission. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru
Malone, D and Murray, H. (2016). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/
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