2 Mechanics Questions

[Thomas154321]

1) Three particles, A, B and C , of masses m, km and 3m respectively, are initially at rest lying in a straight line on a smooth horizontal surface. Then A is projected towards B at speed u. After the collision, B collides with C. The coefficient of restitution between A and B is 1/2 and the coefficient of restitution between B and C is 1/4.

i) Find the range of values of k for which A and B collide for a second time.

ii) Given that k = 1 and that B and C are initially a distance d apart, show that the time that elapses between the two collisions of A and B is 60d/13u.




2) A projectile of unit mass is fired in a northerly directionfrom a point O on a horizontal plain at speed u and an angle θ above the horizontal. It lands at a point A on the plain. In flight, the profectile experiences two forces: gravity, of magnitude g; and a horizontal force of constant magnitude f due to a wind blowing from North to South.

i) Derive an expression, in terms of u, g, f and θ for the distance OA.

ii) Determine the acute angle α such that, for any acute angle θ with θ > α, the wind starts to blow the orifectile back towards O before it lands at A.

iii) An identical projectile, which experiences the same forces, is fired from O in a northerly direction at a speed u and angle 45 degrees above the horizontal and lands at a point B on the plain. Given that θ is chosen to maximise OA, show that OB/OA = (g - f)/((g^2 + f^2)^(1/2) - f).

Describe carefully the motion of the second projectile when f = g.







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