M2 Collisions questions...

Hey, could anyone tell me how to do these couple of questions please:

Ex 4B Q5

A small smooth sphere of mass 3kg moving on a smooth horizontal plane with a speed of 8ms^-1 collides directly with a sphere of mass 12kg which is at rest. Given that the spheres move in opposite directions after the collision, obtain the inequality satisfied by e.

Q8

A sphere of mass m1 moving with speed u1 collides directly with a similar sphere of mass u2 moving with speed u2 in the same direction (u1>u2). The coefficient of restituion between the two spheres is e. Show that the loss of kinetic energy E due to the collision satisfies the equation

2(m1 + m2)E = m1m2(u1-u2)^2 (1-e^2)

Thanks!

Reply 1

uk6458

ok heres what i did for q5:

momentum before = momentum after

(3m x 8) + 0 = (3m x -V) + (12m x Z)

24m = 12mZ - 3mV ...........................................1

restitution:

8e = Z - V

rearrange:

8e +V = Z..........................2

sub 2 into 1:

24m = 12m(8e + V) - 3mV

24m - 96me = 9mV

m's cancel, and factorize:

8/3(1-4e) = V

now it says that the particles go in opposite directions. I have taken left to right to be positive, so that means that, V will go in the negative direction; because it reverses in direction. This means that it V is lesss than 0 :

V < 0

the equation for V is:

8/3(1-4e) = V so 8/3(1-4e) < 0

as v is less than zero, we can get rid of the 8/3 in front, so we get:

1-4e < 0

rearrange:

1 < 4e

divide:

1/4 < e

i hope that is right, could someone please check thanks. :redface:

i hate these inequality questions, you just gotta use the info in the q when it says they go in opposite directions, apply this to the usual restitution and momentum equations, and it seem a bit more resonable.....i think, anyway :redface:

does that help..........?? :smile:

i will attach a diagram in a min.......

Reply 2

Troyes

WOW. I would not have done that if that had come up in the exam. Thanks so much uk6458. God, I hate collisions.

Reply 3

uk6458

thats ok, Troyes, i usually end up in a complete mess with loads of simultaneous equations everywhere on these types of questions......i am suprised i managed this one.....i hate those questions :rolleyes: