The physics books inform us that when two objects with given masses collide, kinetic energy is always conserved.

This appears to follow the common sense rule that energy is always conserved whichever form it is found in.

However we are also told that when two objects collide, such as metal balls for

example, momentum is also conserved. If the two balls are of equal mass then:

mv(1) + mv(2) = mv(3) + mv(4)

This equation would balance under the following circumstances: where m = 1kg, v(1) = 5, v(2) = 1, v(3) = 3 and v(4) = 3. Or in other words:

(1 x 5) + (1 x 1) = (1 x 3) + (1 x 3)

5 + 1 = 3 + 3

However if kinetic energy is also to be conserved (K.E. = ½ mv2) then the following equation has also to balance:

½ mv(1)2 + ½ mv(2)2 = ½ mv(3)2 + ½ mv(4)2

Using the exact same values for the variables as used above for conservation of momentum, the equation determining the conservation of kinetic energy DOES NOT balance:

(1/2 x 1 x 5 x 5) + (1/2 x 1 x 1 x 1) = (1/2 x 1 x 3 x 3) + (1/2 x 1 x 3 x 3)

(12.5) + (0.5) = (4.5) + (4.5)

13 = 9 ????????????

What exactly is wrong here?

How can both momentum and kinetic energy be conserved at the same time?