Linear momentum and its conservation

Linear momentum is a measure of the motion of an object and depends on both its mass and velocity. It is a vector quantity, meaning it has both magnitude and direction.

A minimalist vector diagram showing a small circle (object) with a bold arrow labeled 'v' (velocity) pointing to the right, and a second arrow labeled 'p = mv' (momentum) in the same direction. Keep the design clean and modern.

Definition

Linear momentum (p⃗\vec{p}) is defined as the product of an object's mass (mm) and its velocity (v⃗\vec{v}):

p⃗=mv⃗\vec{p} = m\vec{v}

The SI unit of momentum is kilogram metre per second (kg m s−1^{-1}).

Principle of Conservation of Momentum

The principle of conservation of momentum states that the total linear momentum of a system of interacting particles remains constant, provided no external resultant force acts on the system.

Important

Momentum is always conserved in any interaction or collision, as long as the system is isolated from external forces.

Mathematically, for two objects before and after a collision:

m1u1+m2u2=m1v1+m2v2m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

where u1u_1 and u2u_2 are initial velocities, and v1v_1 and v2v_2 are final velocities.

A clean, side-by-side illustration of two blocks (labeled m1 and m2) moving towards each other on a straight line, each with arrows showing their initial velocities (u1 and u2). Below, show the same blocks after collision with arrows for final velocities (v1 and v2). Use minimal color and simple shapes.

Types of Collisions

  • Elastic collision: Both momentum and kinetic energy are conserved.
  • Inelastic collision: Momentum is conserved, but kinetic energy is not (some is converted to other forms like heat or sound).
  • Perfectly inelastic collision: The colliding objects stick together after the collision.

Minimalist illustration showing two blocks colliding: first panel, blocks approach; second panel, blocks stick together and move as one after collision (perfectly inelastic). Use simple shapes and arrows to indicate motion.

Kinetic Energy in Collisions

  • In elastic collisions, total kinetic energy before and after the collision is the same.
  • In inelastic collisions, some kinetic energy is lost.
Definition

In an elastic collision, the relative speed of approach equals the relative speed of separation:

∣u1−u2∣=∣v2−v1∣|u_1 - u_2| = |v_2 - v_1|

Momentum Conservation in Two Dimensions

When objects collide at angles, momentum is conserved in both the xx and yy directions. You must resolve velocities into components and apply conservation of momentum separately in each direction.

Minimalist diagram of two circles colliding at an angle. Show initial velocity vectors for both, and after collision, show velocity vectors at new angles. Overlay faint x and y axes to indicate component directions.

Worked Example

1

Example

Two trolleys of mass 2 kg and 3 kg move towards each other at 4 m s−1^{-1} and 2 m s−1^{-1} respectively. They collide and stick together. Find their common velocity after the collision.

Key Points

  • Momentum is always conserved in collisions and explosions if no external force acts.
  • Kinetic energy may not be conserved (except in elastic collisions).
  • Conservation of momentum applies in all directions (vector quantity).
  • Use the correct sign for velocity (direction matters).
Exam Tip

Always state the principle of conservation of momentum before using it in calculations. Show all steps and include units.

Summary:

  • Momentum: p⃗=mv⃗\vec{p} = m\vec{v} (vector, units: kg m s−1^{-1})
  • Conservation: Total momentum before = total momentum after (if no external force)
  • Elastic collision: Both momentum and kinetic energy conserved
  • Inelastic collision: Only momentum conserved
  • Apply conservation separately in each direction for 2D problems
← Non-uniform motion

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