A0.105-kg soccer ball moving at 24 m/s is caught and held by a 75-kg goalie at rest. with what speed does the goalie fly backward? a. 0.026 m/s c. 0.340 m/s b. 0.034 m/s d. 1.03 m/s

When it comes to momentum, the total momentum before collision is conserved, so that means the momentum after collision should be the same. We solve momentum with this formula:



where:
m1 = mass of object 1
m2 = mass of object 2
v1 = velocity of object 1 before collision
v2 = velocity of object 2 before collision
v'1 = velocity of object 1 after collision
v'2 = velocity of object 2 after collision

In your case you are looking for v'2 because you are looking for the speed of the goalie after he caught the ball. 

To give you a better view on this, let's see what your given are:
m1 = 0.105 kg (soccer ball mass)
m2 = 75 kg (Goalie mass)
v1 = 24 m/s (how fast the ball was going before it was caught)
v2 = 0 m/s (The goalie was not moving or at rest)
v'1 = 0m/s (The goalie held the ball)
v'2 = ? 

Now we just input all of these data into our equation and solve for what we do not know which is v'2.




The next step is to do all the operations that you can first before we weed out the missing value:




Now we can solve for the missing value by transposing all other elements to the opposite side of the equation, which leaves the missing value on one side alone. Now remember when you transfer sides, you do the opposite operation. Now you multiply 75kg in one side, so when you transfer it it will come out as division:




If you round it off the answer would be 0.034m/s