Max Value of Coefficient of Restitution for 3 Sphere Collision

[markosheehan]

three spheres of mass 1kg,2kg and 3kg move in the same line with velocities 5i,i and 3i respectively. the smaller masses are the first to collide.if only one collision takes place find the maximum value for the coefficient of restitution between the smaller masses.

what I've got so far: 5(1)+1(2)=1(x)=2(y) using the law of conservation of momentum. 7=x+2y

using law if restitution -e=x-y/5-1 -4e=x=y

i also know y>x as there no more collisions. y>3 also if there are no more collisions.

 

[jvicens]

Firstly, great job using the law of conservation of momentum to solve for the initial velocities of the masses. This is a crucial step in understanding the motion of the system.

To find the maximum value for the coefficient of restitution, we need to consider the elastic collision between the smaller masses. In an elastic collision, the coefficient of restitution (e) is defined as the ratio of the relative velocity of separation to the relative velocity of approach. In this case, the relative velocity of approach is 5i (since the smaller masses are moving towards each other) and the relative velocity of separation is 3i (since the smaller masses are moving away from each other after the collision).

So, we can write the equation for the coefficient of restitution as:

e = (relative velocity of separation)/(relative velocity of approach)

e = (3i)/(5i)

e = 0.6

Therefore, the maximum value for the coefficient of restitution between the smaller masses is 0.6. This means that after the collision, the smaller masses will move away from each other with 60% of their initial relative velocity.

It is important to note that this is the maximum value for the coefficient of restitution because any value higher than 0.6 would result in the smaller masses moving away from each other with a relative velocity greater than 3i, which would lead to another collision. This is not possible since we have already established that there will be no more collisions after the initial one.

I hope this helps in understanding the concept of coefficient of restitution and its application in this scenario. Keep up the good work as a scientist!