How Do Velocities Change After a Collision Between Two Spherical Particles?

[ly667]

A smooth spherical particle with mass 2.5 kg collides with a second smooth spherical
particle of mass 6kg. Before the collision, the first particle has velocity 9ms^1
in the positive x-direction while the second particle has velocity 6ms^1 in the positive y-direction. At the instant of collision, the particles are so that the normals to their surfaces at the point of contact are ±n, where n
is a unit vector inclined at -pi/4 from the x-axis as shown. You may assume that the coefficient of restitution e = 2/3 and that no other forces are involved. Determine the velocities v1,1 and v2,1 of the first and second particles respectively, immediately after the collision.

 

[tiny-tim]

welcome to pf!

hi ly667! welcome to pf!

show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[ly667]

I don't really have much of an idea to be honest, nor do my classmates. We have been looking at it all night :(
Basically what I have so far is.. I have constructed my diagram.
Particle 1: m1=2.5, v1=9
Particle 2: m2=6, v2=6

Particles interact along line, to n=v1-v2
we also need m, a vector perpendicular to n
m=v1+v2 (so m.n=0)

Physical assumptions: momentum conservation
coeff of rest 2/3
smooth collisions

 

[tiny-tim]

this is an inelastic collision

so you need conservation of momentum in two perpendicular directions (either x and y, or n and z×n)

and one more equation, which will be the coefficient of restitution equation

(the coefficient of restitution is the ratio of the relative speed after to the relative speed before the collision, see the pf library)

show us what you get ​

 

[ly667]

Okay, so far I have

Mathematically condition i.) gives
m1,v1,0+m2v2,0=m1,v1,1+m2v2,1

ie. 2.5(v1,1)+6(v2,1)=22.5+36 = 58.5

while ii.) gives(v2,1-v1,1)= e(v2,0-v1,0).n

and iii.) gives (v2,1-v1,1).m= (v2,0-v1,0).m

=0 since n=v2,0-v1,0

Its easiest to work in components

let v2,1=9u2+6w2
and v1,1=9u1+6w1

momentum cons becomes

2.5(9u1)+2.5(6w1)+6(9u2)+6(6w2)=58.5

I am stuck and don't know what to do from here.. Is what I have done correct?

 

[tiny-tim]

ie. 2.5(v1,1)+6(v2,1)=22.5+36 = 58.5

ah, no wonder you're stuck, your professor obviously hasn't explained to you the vector nature of conservation of momentum

your 58.5 is the sum of the x and y components of momentum …

you can't do that!

momentum is a vector, and obeys the laws of vector addition

you can't add components in different directions

write out two equations, one for conservation of momentum in the x direction, and one for conservation of momentum in the y direction ​

 

[ly667]

So should I use my velocity value as vectors, ie instead of v1=9, let it equal 9i?

 

[tiny-tim]

yes

 

[ly667]

Okay :) thank you! Will do that now! Could you please help me with my collisions question? I have gotten a good bit of it done but am stuck at a point!