Conservation of momentum
Newton's Laws & Momentum - OCR A-Level Physics
Key Definition
Principle of conservation of momentum
The total momentum of a system of objects stays constant, as long as no net external force acts on the system. A system with this property is called a closed systemA system where no external resultant force acts, so total momentum is conserved..
The total momentum of a system of objects stays constant, as long as no net external force acts on the system. A system with this property is called a closed systemA system where no external resultant force acts, so total momentum is conserved..
Case 1 : Collision without sticking
Two objects approach, interact, and separate. Use:
$$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$
Case 2 : Objects stick together
After the collision the two objects move as one combined mass with one velocity:
$$m_1 u_1 + m_2 u_2 = (m_1 + m_2) v$$
Case 3 : Explosion or separation
A single object splits into two (or more) pieces. If initially at rest, total momentum is zero before and after:
$$m u = m_1 v_1 + m_2 v_2$$
- Conservation applies to all interactions: collisions, explosions, separations, rebounds and recoils.
- It follows from Newton's third lawIf A pushes B, B pushes A with an equal and opposite force of the same type.: equal and opposite forces act on the two objects for the same time, producing equal and opposite changes in momentum that cancel.
- "Closed system" means no external resultant force. Internal forces (between objects inside the system) do not break conservation.
- Choose a positive direction at the start. A 3.0 kg trolley moving right at 4.0 $\text{m s}^{-1}$ has $p = +12 \; \text{kg m s}^{-1}$. The same trolley moving left has $p = -12 \; \text{kg m s}^{-1}$.
Common Mistake
HIGH
Wrong: Forgetting the sign of velocity when two objects move in opposite directions, so the "conserved" momentum comes out wrong.
Right: Define a positive direction first. If A moves right at $+5.0 \; \text{m s}^{-1}$ and B moves left at $3.0 \; \text{m s}^{-1}$, write B's velocity as $-3.0 \; \text{m s}^{-1}$ in the equation. Keep the same sign convention all the way through.
Right: Define a positive direction first. If A moves right at $+5.0 \; \text{m s}^{-1}$ and B moves left at $3.0 \; \text{m s}^{-1}$, write B's velocity as $-3.0 \; \text{m s}^{-1}$ in the equation. Keep the same sign convention all the way through.
Examiner Tips and Tricks
- Write the conservation equation with the SAME LHS variables every time: $m_1 u_1 + m_2 u_2 = \dots$. Then choose Case 1, 2, or 3 for the RHS depending on the question.
- When stating the principle, include the condition "as long as no resultant external force acts" - it is worth a mark on its own.
- A negative answer for a velocity is not a mistake. It means the object moves in the direction you chose as negative.
Before/after sketch of a head-on collision between two trolleys: arrows showing $u_1, u_2$ before and $v_1, v_2$ after. Positive direction marked with a "+x" arrow.