EXAMPLES. 313 



17. A particle, describing an ellipse about a focus, strikes a fixed plane 

 through the focus at right angles to the major axis. Prove that, if the 

 coefficient of restitution is equal to the eccentricity, the major axis of the 

 new orbit is half that of the old. 



18. Prove that the impulse necessary to make a particle of unit mass, 

 moving in an equiangular spiral of angle a under the action of a force to the 

 pole, describe a circle under the action of the same force, is 



r being the distance from the pole, and F the force at the moment of impact. 



19. A particle is describing an ellipse of eccentricity e about a focus and 

 when its radius vector is half the latus rectum it receives a blow which makes 

 it move towards the other focus with a momentum equal to that of the blow. 

 Find the position of the axis of the new orbit and show that its eccentricity 



20. A particle of mass m is projected from a point P with velocity V and 

 moves under a force to a fixed point S varying inversely as the square of the 

 distance. PP is the chord through the other focus of the path. When the 

 particle reaches P' the kinetic energy is increased by ^wF 2 //(4a R) by a 

 tangential impulse, R being the distance SP and 2a the major axis of the 

 orbit. Prove that the new path is independent of the direction of projection. 



21. A comet describing a parabola of latus rectum 21 before it has 

 reached the apse collides at a point of its orbit distant R from the Sun 

 with another comet of equal mass falling from rest at an infinite distance 

 directly towards the Sun, and the two comets coalesce. Prove that the 

 subsequent orbit is an ellipse of major axis 2a, given by (l-l/R) 2 = < 2R[a. 



22. A particle is describing an ellipse about a centre of force in one focus 

 S t and when it is at the end E of the further latus rectum it receives a blow 

 in direction SE which makes it move at right angles to SE. Find the 

 momentum generated by the blow, and prove that the particle will proceed 

 to describe an ellipse of eccentricity x /{2e 2 /(l +e 2 )}. 



23. A particle is describing an ellipse about a focus S, and when it is at 

 one end of the latus rectum it receives a blow which makes it describe a 

 confocal hyperbola. Prove that the direction of the blow makes with the 

 tangent to the ellipse an angle cot" 1 ^, where e is the eccentricity of the 

 ellipse. 



24. A shell of mass M is moving with velocity 1 *. An internal explosion 

 generates an amount E of energy and thereby breaks the shell into two 

 fragments whose masses are in the ratio m 1 : m z . The fragments continue 

 to move in the original line of motion of the shell. Prove that their 

 velocities are 



V+ ^amgJS/mjJf), V- x /(2m 1J '/wi. 2 M ). 



