Two spheres travelling in opposite directions collide

[Woolyabyss]

Homework Statement

Two spheres of radius r move horizontally in opposite directions. The first has mass 2m and speed 2u, the second has mass m and speed 4u. The coefficient of restitution is (1/√3). The centres of the two spheres lie on two parallel lines, a distance r apart.

(i) show that at the moment of impact, the line of centres of the spheres make an angle of 30° with their previous lines of motion.

(ii) find the speeds of the spheres after impact.

Homework Equations

conservation of momentum and coefficient of restitution

The Attempt at a Solution

the i axis is along the line of their centres at impact.
the j axis is vertical to the i axis.

∅ is angle with which the line of the centres of the spheres makes an angle with there previous lines of motion.(they are parallel)

there is no change in j.

(i)

conservation of momentum along the i axis.

2m(2ucos∅ ) + m(-4ucos∅) = 2m(v1) + m(v2)

0 = 2v1 + v2

v2 = -2v1

coefficient of restitution

( v1 -(-2v1) )/(2ucos∅ -( -4ucos∅ ) ) = 3v1/6ucos∅ = - 1/√3

simplify ...

v1 = (-2/√3)ucos∅

Im not exactly sure what to do from here I know that cosinverse(√3/2) = 30

so presumably I should be able to find v1 = u and cancel them.

Im guessing it has something to do with their centers being a distance r apart but I'm not exactly sure what to do with that information.
Any help would be appreciated.

 

[TSny]

Question (i) is purely a geometry or trig question. No physics needed. Draw a diagram of the spheres at impact.

 

[Woolyabyss]

Question (i) is purely a geometry or trig question. No physics needed. Draw a diagram of the spheres at impact.

I drew a diagram of the two spheres at impact like you said. using the line between the centre of the spheres and the r distance between there centres i constructed a right angled triangle.I was able to use trig to figure out the angle was 30°.

for (ii) I subsituted the value of cos30 into.... v1 = (-2/√3)ucos∅

and figured out v1 = -u I Substituted this into the equation v2 = -2v1 which means v2 = 2u

since the j vector is the same before and after I used Pythagoras theorem for both spheres and figured out the j vector.

I then used Pythagoras theorem again and figured out there speeds after impact(the resultant vector of both the j and i components) were u√2 and u2√2

which according to the back of my book are the right answers so thanks for the help.

 

[TSny]

Good work.

 

[Nickie]

Your solution for part (i) is correct. To continue, you can use the Pythagorean theorem to find the distance between the centers of the spheres after impact. This distance will be r, since the centers were initially r apart and they will still be moving along parallel lines. Then, you can use the angle of 30° and the distance r to find the speeds of the spheres after impact. You can use the conservation of energy equation, as well as the coefficient of restitution, to solve for the final speeds.