Conservation of linear momentum

The principle that the total momentum of a system is unchanged by a direct collision — how to set it up with a consistent positive direction, the special case where the bodies coalesce (stick together) and move with a common velocity, and how to read the sign of the answer as a direction.

Mechanics · Momentum

Conservation of linear momentum

Why it works

When two bodies collide directly (moving along the same straight line, hitting head-on or one catching the other), the forces they exert on each other are equal and opposite, and they act for the same length of time. The effect on the total momentum of the two bodies cancels out. So, provided no other external force acts along the line during the impact,

\textbf{total momentum before} = \textbf{total momentum after}.

This is the principle of conservation of linear momentum. For two bodies:

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2,

where

u

are the velocities just before and

v

the velocities just after.

Set a positive direction and stick to it. Every velocity in the equation — before and after, for both bodies — is measured with the same sign convention. A body moving the "negative" way goes in with a minus sign. This is the step that goes wrong most often: if two particles approach each other, one of the "before" velocities must be negative.

Coalescing bodies. If the two bodies stick together on impact (coalesce), they move off as a single object of mass

m_1 + m_2

with one common velocity

v

m_1 u_1 + m_2 u_2 = (m_1 + m_2)\,v.

Use the combined mass on the right — this is the commonest case examined, and the commonest place to slip by dividing by only one of the masses.

Reading the answer. Solve for the unknown as an ordinary signed number. If a velocity comes out negative, the body is moving in the negative direction — for a struck body that often means it has reversed, or that your chosen positive direction was the other way. State the direction, don't just quote a magnitude.

Momentum is conserved in the collision; kinetic energy generally is not (some is lost to sound, heat and deformation, especially when bodies coalesce). Never assume speeds are unchanged or that they simply swap.