Direct Collisions with objects moving in towards each other

[Woolyabyss]

Homework Statement

Two Smooth spheres of masses m and 3m move with speeds u and v in opposite directions. The smaller mass is brought to rest.

(i) show that the coefficient of restitution,e, is given by e =(u - 3v)/(3u + 3v)

(ii) show that u ≥ 3v

Homework Equations

m1u1 + m2u2 = m1v1 + m2v2

(v1 - v2)/(u1 - u2) = -e

The Attempt at a Solution

(i)
let x equal speed of bigger object after impact

3mv + m(-u) = 3mx

x = (3v -u)/3

using (v1 - v2)/(u1 - u2) = -e


x/(v -(-u) ) = -e

substitute value for x

(3v -u)/(3v + 3u) = -e

e = (u - 3v)/(3v + 3u)

(ii)

can somebody help with this part? I have no Idea how to turn this into an inequality.

Any help would be appreciated

 

[Simon Bridge]

3mv + m(-u) = 3mx

... you have the small sphere traveling in the negative direction before impact and then stopping.
This is a head-on collision: which direction must the larger sphere be traveling after the collision?
What does the speed of the small mass have to have been in order for this to happen?

Now take a look at the equation for the coefficient of restitution ... what happens to it when u < 3v?

 

[Woolyabyss]

... you have the small sphere traveling in the negative direction before impact and then stopping.
This is a head-on collision: which direction must the larger sphere be traveling after the collision?
What does the speed of the small mass have to have been in order for this to happen?

Now take a look at the equation for the coefficient of restitution ... what happens to it when u < 3v?

The larger sphere has to travel in the opposite(negative) direction.

When u < 3v the coefficient of restitution will be negative. (I'm not sure what this means for the impact)

I'm not sure how to work out what speed the small has to be.

 

[Simon Bridge]

I'm not sure how to work out what speed the small has to be.

Hint: conservation of momentum.

 

[Woolyabyss]

Hint: conservation of momentum.

Is this correct?

3v - u = 3x

does it have to be u ≥ 3v since we know the bigger mass has to travel in the opposite direction after impact.

which means its magnitude is a negative value. otherwise the bigger mass would continue in the same direction.Which we know can't happen since the smaller one stops.
eg. u = 3.2v 3v - 3.2v = -.2v = 3x

but it could also be equal since they could both stop when they collide.
3v - 3v = 0

hence ......u ≥ 3v

 

[Simon Bridge]

does it have to be u ≥ 3v since we know the bigger mass has to travel in the opposite direction after impact.

That's the right idea - I'll just help you tidy up the math a bit:

So - by conservation of momentum: 3v-u=3x => u=3(v-x) (note: u,v>0)
If the small ball stops, the big ball cannot continue forward (the small ball is in the way), so it must stop or go backwards, so x≤0
For x=0, u=?
For x<0, u=?

This can also be obtained from the physical meaning of that e value ... you should take another look at your notes there.

 

[Woolyabyss]

That's the right idea - I'll just help you tidy up the math a bit:

So - by conservation of momentum: 3v-u=3x => u=3(v-x) (note: u,v>0)
If the small ball stops, the big ball cannot continue forward (the small ball is in the way), so it must stop or go backwards, so x≤0
For x=0, u=?
For x<0, u=?

This can also be obtained from the physical meaning of that e value ... you should take another look at your notes there.

Alright thanks

 

[Woolyabyss]

That's the right idea - I'll just help you tidy up the math a bit:

So - by conservation of momentum: 3v-u=3x => u=3(v-x) (note: u,v>0)
If the small ball stops, the big ball cannot continue forward (the small ball is in the way), so it must stop or go backwards, so x≤0
For x=0, u=?
For x<0, u=?

This can also be obtained from the physical meaning of that e value ... you should take another look at your notes there.

x = 0, u = 3v

say x= -1, u =3(v - 1) = 3v - 3 would this not mean ... u < 3v?

or is it because you brought the 3x so it changes from negative to positive
so it would actually be... 3v + 3

and so u > 3v ?