If there is no external force, the total momentum of a system remains constant. This is a universal law applicable to collisions, explosions, splitting, and many other phenomena.
Basic Knowledge
Law of conservation of momentum: When the net external force acting on a system is zero, the sum of the momenta of the entire system is conserved.m₁v1 + m₂v2 = m₁v1 + m₂v2 (before and after collision)
(v1, v2: velocities before collision, v1, v2: velocities after collision)
Derivation: By the third law, the force exerted by A on B and the force exerted by B on A are equal in magnitude and opposite in direction. Their impulses cancel each other, so the change in the total momentum of the system is zero.
For a two-dimensional collision, set up conservation equations separately for the x and y components:
x component: m₁v1ₓ + m₂v2ₓ = m₁v1ₓ' + m₂v2ₓ'
y component: m₁v1ᵧ + m₂v2ᵧ = m₁v1ᵧ' + m₂v2ᵧ'
Conservation law: m₁v1 + m₂v2 = m₁v1 + m₂v2 (in a straight line)
Condition: net external force = 0 (internal forces only)
Internal force: force acting between objects within a system (an action–reaction pair)
Explosion or splitting: initially at rest (total momentum = 0) → sum remains 0 after splitting
Two dimensions: conserve each x and y component independently
Further Study (Background and Significance)
The law of conservation of momentum is a highly fundamental law derived directly from Newton's third law. Even when external forces exist, conservation holds approximately in collisions of extremely short duration, when the impulse can be neglected.
In explosion and splitting problems, if the object is initially at rest, the total momentum is zero. After the explosion, the two objects have equal-magnitude, oppositely directed momenta: m₁v1 = −m₂v2
Recoil (rocket propulsion): when a rocket ejects gas backward, conservation of momentum accelerates the rocket forward. In outer space, there is no external force, so the conservation law holds purely.
In a “smooth horizontal surface” setting, friction is absent and vertical forces are canceled by the floor, so the total horizontal external force is zero → horizontal momentum is conserved.
- Net external force = 0 → momentum is conserved
- Internal forces (action–reaction) do not change the total momentum of a system
- Initially at rest before an explosion → total momentum after the explosion = 0
- “Smooth horizontal surface” → conservation in the horizontal direction
- Two dimensions → set up equations using x and y components
- Conservation of momentum is a vector equation (pay attention to signs)
- Even if F ≠ 0, momentum is approximately conserved when Δt is extremely small
Points to Note (Common Confusions)
① Conservation of momentum and conservation of energy are different laws. Momentum is always conserved, but kinetic energy decreases in an inelastic collision. ② Conservation applies only in directions with zero external force. For example, if gravity acts vertically, conservation may apply only horizontally. ③ Momentum is a vector. Calculating with “speed” can cause signs to be overlooked. ④ In a collision where objects stick together, v1 = v2 = V (a common velocity), simplifying the equation.
Practice
- An object of mass 2.0 kg moving at 3.0 m/s collides with and sticks to a stationary object of mass 3.0 kg. Find their velocity after the collision.
- A stationary object of mass 4.0 kg explodes, and a 2.0 kg piece flies to the right at 6.0 m/s. Find the velocity of the remaining piece.
- State the condition under which the law of conservation of momentum holds.