Do We Include Signs in Conservation of Momentum Equations?

[Mr Davis 97]

When we use the conservation of momentum with, for example, collisions do we include the sign with the velocities or are the signs inherent in the quantity? For examples, would we write ##m_1v_1 = m_1v_1 + m_2v_2## or ##m_1v_1 = -m_1v_1 + m_2v_2## for a collision where a moving object hits a stationary one and then moves backwards while the stationary object moves forward.

 

[blue_leaf77]

Momenta are vector quantities, so direction matters. In your problem, assume the event lies in the x coordinate. What are the initial and final momentum vectors of the object?

 

[Nugatory]

You always add the momenta; conservation of momentum for a two-body straight-line collision says that ##m_1v_1+m_2v_2=m_1v_1'+m_2v_2'## and it's always a plus sign. (##m_1## is the mass of the first object and ##v_1## and ##v_1'## are its velocities before and after the collision; similarly for the second object with mass ##m_2##).

However, the momentum itself will be a positive or negative number depending on whether the velocity is positive or negative. For example, if we collide a moving object of mass ##m_1=3## kg with a stationary object of mass ##m_2=6## kg, we start with ##v_1## positive and ##v_2## equal to zero. After the collision ##v_1'## is negative and ##v_2'## is positive.

It would be a good exercise to try three different examples: one in which ##m_1## is less than ##{m}_2## (as above), one in which they are equal, and one in which ##m_1## is greater.

 

[Chestermiller]

Since momentum is a vector quantity, if you write the momentum balances with the unit vectors included, you can never go wrong.

 

[Biker]

I think he means if for example the magnitude of ##V_1## is ##V_1## and the direction is for example set to negative. He is asking whether he should substitute the magnitude with a negative sign.

You could keep it as a vector quantity until the end of your equation and then substitute the negative sign with the magnitude of the velocity or substitute it from the beginning it is up to you. Think of it as If it is a vector you don't need negative and positive. If you change it to magnitudes then you have to express opposite vectors by giving one of the vectors a negative signs

 

[sophiecentaur]

you can never go wrong.

Wanna bet?

 

[Chestermiller]

Wanna bet?

Ha! Good one.

Chet