Max (15 kg) and maya (12 kg) are ice-skating on a frozen pond. at one moment, when max is skating away from the shore at 8.2 m/s and maya is skating towards the shore at 4.6 m/s, they collide and bounce off each other without falling. if maya rebounds at 3.4 m/s, what is max’s velocity after their collision?

Using the Law of Conservation of Momentum, we can solve for the velocity of Max after collision with this formula:

 m1v1i  +  m2v2i      =      m1v1f + m2v2f

Total Momentum             Total Momentum
 Before collision                 After collision

Where: m1 = mass of first object (Max)
             m2 = mass of second object (Maya)
             v1i = velocity of first object before collision (Max)
             v1f = velocity of first object after collision (Max)
             v2i = velocity of second object before collision (Maya)
             V2f = velocity of second object after collision (Maya)

This means that the total momentum before the collision is sustained even after collision.

First we sort out what you know:
m1 = 15 Kg (This is Max)
m2 = 12 Kg (This is Maya)
v1i = 8.2 m/s
v2i = -4.6 m/s 
v2f = 3.4 m/s
v1f = ?

Now you will notice that Maya has a negative velocity before collision, this is because she is coming from the opposite direction. Her velocity after collision is positive because she rebounded, which means she went backwards, following the direction Max was coming from.

Now using the formula we will input the data that we have:

(15kg)(8.2m/s) + (12kg)(-4.6m/s) = (15Kg)(v1f+ (12kg)(3.4m/s)

It is very important that you know the direction of the object because the sign matters. Now let's simplify the equation:

123kg.m/s + (-55.2kg.m/s) = (15kg)(v1f) + 40.8 kg.m/s
                        67.8kg.m/s =  (15kg)(v1f) + 40.8 kg.m/s

To find the unknown, we must isolate it in one side of the equation so we transpose our values from the right side to the other, following the rule that when you transpose you change the operation used to its opposite. Hence, addition is to subtraction; multiplication is to division.

The formula will then look like this:

 = v1f

Then we simplify the equation from there:

= v1f

We cancel out the kg and we are left with m/s:
1.8m/s = v1f

The velocity of Max after collision is 1.8m/s. 

The answer is positive, just like his velocity before collision, so this means he is going the same direction as he was before they collided.