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.







Hope you enjoy these.

[VBAhack]

Good problems. I especially like that the result is included in the problem statement so you can tell if you did it right. Sounds like you are in a great class. Challenging problems with simple results. I found the 1st a bit more difficult than the 2nd.

Since you didn't ask any questions, I presume you were able to solve them?

[Thomas154321]

I posted them before I properly looked at them, so anyone could if they wanted. I have since solved the first one and most of the second one. I found the second one harder actually, don't know why. The first one was tidy.

[VBAhack]

I have since solved the first one and most of the second one.

Terrific! Actually, I had to do them both twice because I goofed on the 1st try. Keep them coming!

[Thomas154321]

I haven't been asked to solve this one yet, but here's the only other one I currently posess. I won't be solving this until I'm asked:





A painter of weight kW uses a ladder to reach the guttering opn the outside wall of a house. The wall ia vertical and the ground is horizontal. The ladder is modelled as a uniform rod of weight W and length 6a.

The ladder is not long enough, so the painter stands on the ladder on a uniform table. The table consists of a square of side a/2 with a leg of length a at each corner. The weight of the table is 2W. The foot of the ladder is at the centre of the table top and the ladder is inclined at an angle arctan 2 to the horizontal. The edge of the table nearest the wall is parallel to the wall.

The coefficient oif friction between the foot of the ladder and the table top is 1/2. The contact between the ladder and the wall is sufficiently smooth for thr effects of friction to be ignored.

a) Show that, if the legs of the table are fixed to the ground, the ladder does not slip on the table however high the painter stands on the ladder.

b) It is given that k = 9 and that the coefficient of friction between each table leg and the ground is 1/3. If the legs of the table are not fixed to the ground, so that the table can tilt or slip, determine which occurs first when the painter slowly climbs the ladder.


Have fun!

[Rich2189]

mechanics questions are the best!, if you want a good read, look for some perm and coms questions in statistics, they read like mud.