1.14 Conservation of Linear Momentum

• The principle of conservation of momentum states that in a closed system, the total momentum before an event is equal to the total momentum after the event

• Momentum is always conserved in collisions where no external forces act

• This is usually written as:

Total momentum before a collision = Total momentum after a collision

• Since momentum is a vector quantity, a system of objects moving in opposite directions can have an overall momentum of 0
  • This applies to objects moving towards each other or away from each other

• The diagram below shows two masses m with velocity u and M at rest (M has zero velocity)

The momentum of a system before and after a collision

• Before the collision:
  • The momentum is only of mass m which is moving
  • If the right is taken as the positive direction, the total momentum of the system is m × u

• After the collision:
  • Mass M also now has momentum
  • The velocity of m is now -v (since it is now travelling to the left) and the velocity of M is V
  • The total momentum is now the momentum of M + momentum of m
  • This is (M × V) + (m × -v) or (M × V) – (m × v)

• Newton’s third law of motion states:

Whenever two bodies interact, the forces they exert on each other are equal and opposite

• This means:
  • When one object exerts a force on another object, the second object will exert an equal force on the first object in the opposite direction
  • When two objects collide, both objects will react, generally causing one object to speed up (gain momentum) and the other object to slow down (lose momentum)

Newton's third law can be applied to collisions

• Consider the collision between two trolleys, A and B:
  • When trolley A exerts a force on trolley B, trolley B will exert an equal force on trolley A in the opposite direction

• In this case:

F_B–A = –F_A–B

• While the forces are equal in magnitude and opposite in direction, the accelerations of the objects are not necessarily equal in magnitude
• From Newton's second law, acceleration depends upon both force and mass, this means:
  • For objects of equal mass, they will have equal accelerations
  • For objects of unequal mass, they will have unequal accelerations