46, 47] PARABOLIC MOTION. 47 



of projection, then the parabolic path for which the range on a line through 8 

 is greatest touches this parabola at the point where the line cuts it. 



[From this it follows that all possible paths of particles moving with 

 uniform acceleration g downwards, and starting from a point S with given 

 velocity V, touch a paraboloid of revolution about the vertical through $ 

 having its focus at S. This paraboloid is the envelope of the trajectories of 

 such particles.] 



9. If v is the velocity at any point of the parabolic path, and p the radius 

 of curvature at the point, verify that v 2 /p is equal to the component of g along 

 the normal. 



10. Prove that, in the different trajectories possible under gravity 

 between two points A, B, the times of flight are inversely proportional to the 

 velocities when vertically over the middle point of AB. 



11. Two particles describe the same parabola under gravity. Prove that 

 the intersection of the tangents at their positions at any instant describes a 

 coaxial parabola as if under gravity. Prove also that, if r is the interval 

 between the instants when they pass through the vertex, the distance between 

 the vertices of the two parabolas is ^#r 2 . 



12. A particle moves under gravity from the highest point of a sphere of 

 radius c. Prove that it cannot clear the sphere unless its initial velocity 

 exceeds 



13. Prove that there are in general two directions in which a particle 

 may be projected with a given velocity so as to strike a given object, and find 

 how far off in a given direction the object may be. 



14. Prove that the greatest range on an inclined plane through the point 

 of projection is equal to the distance through which the particle would fall 

 during the time of flight. 



47. Simple Harmonic Motion. A point moving in a 

 straight line with an acceleration directed to a point in the line 

 and proportional to the distance is said to have a simple harmonic 

 motion. 



Let the line be the axis x, and the point towards which the 

 acceleration is directed the origin ; then the acceleration is in the 

 negative direction of the axis when x is positive, and in the 

 positive direction when x is negative, so that the formula for 

 it is 



X = fjLX, 



where //, is a positive constant. 



Let time be measured from an instant at which x a and 

 c = 0. With the origin . as centre and with radius a describe a 



