Physics/Proving the law of conservation of momentum
Hi there. How are you? I was shown on TV once how to prove the law of conservation of momentum I.e. momentum before = momentum after. The written proof involved trolleys, steps and explanations and equations. It also used Newton's Third Law of Motion. I wish I still had the darn thing. Would you be able to help me prove it in the same fashion at grade 12 level if its not too much trouble? If possible a hand written proof would do wonders. Its not a homework question. Its just me going back in time to grade 12 and keeping a printed document for my collections. My email address is petergoodwill(at)telkomsa.net if it needs to be hand written. This would be sincerely appreciated.
PS Have a wonderful festive season.
Sorry for the delay in answering. I can do that using this study of trolley A and trolley B. Let them have masses Ma & Mb and original velocities Ua and Ub. When they collide, trolley A exerts average force Fa on trolley B while trolley B exerts average force Fb on trolley A. The time of contact for both trolleys is time t.
Using Newton's 2nd law, we know that trolley A experiences acceleration
aA = Fb/Ma
(Because acceleration is usually a lower case a, I'll make the subscript, designating which trolley's acceleration it is, an upper case.)
That causes the velocity of trolley A to change to
Va = Ua + aA*t = Ua + Fb*t/Ma
Likewise, we know that trolley B experiences acceleration
aB = Fa/Mb
That causes the velocity of trolley B to change to
Vb = Ub + aB*t = Ub + Fa*t/Mb
The total original momentum of the 2 trolleys is Ma*Ua + Mb*Ub.
Their final momentum, after the collision, is Ma*(Ua + Fb*t/Ma) + Mb*(Ub + Fa*t/Mb).
If momentum is conserved, then we must be able to set the original and final momentum expressions equal to each other and show that is truly an equality. So
Ma*Ua + Mb*Ub =? Ma*(Ua + Fb*t/Ma) + Mb*(Ub + Fa*t/Mb)
Expanding the right side
Ma*Ua + Mb*Ub =? Ma*Ua + Ma*Fb*t/Ma + Mb*Ub + Mb*Fa*t/Mb
Transferring the Ma*Ua and Mb*Ub terms from right to left and simplifying the remainder of the right side
0 =? Fb*t + Fa*t
Obeying Newton's 3rd law, we realize that Fb = -Fa. Therefore -Fa can be substituted for Fb giving
0 =? -Fa*t + Fa*t
and indeed, 0 does truly equal -Fa*t + Fa*t
So using the basic kinematic formula Vf = Vo + a*t and Newton's 2nd and 3rd laws, we have shown that expressions for momentum before and after the collision that are equivalent, proving that momentum is conserved.
I hope this helps,