Impulse and Momentum

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Impulse and Momentum is the eighth lecture in the Mechanics section of PH1011. It covers the conservation of momentum, impulse, Newton's second law in context of a stream of particles, and the centre of mass.

Next: Collisions

Motivation
Particles interact via repulsive or attractive forces. This can be considered a collision even if they do not actually touch, and causes the particles to move off at a different angle due to Newton's third law. This gives F(A->B) + F(B->A) = 0; eg m1a1 + m2a2 = 0. Integrating this by time finds the conservation of linear momentum; m1v1 + m2v2 = constant.

This is stated as "In the absence of external forces, the total momentum before any collision is equal to the total momentum after the collision".

Centre of Mass
If the total mass is M = m1 + m2, then the conservation of linear momentum can be given as M(m1v1 + m2v2)/m1+m2 = Mvcom = constant. vcom is the centre of mass velocity, which is a vector of fixed magnitude and direction between the interacting particles. Two other centres of mass - acceleration and position - can be found using differentiation and integration respectively.


 * rcom = 1/M(m1r1 + m2r2)
 * vcom = 1/M(m1v1 + m2v2) = constant
 * acom = 1/M(m1a1 + m2a2) = 1/M(F1 + F2) = 0

The centre of mass in a system can be found by considering an equilibrium of two point masses - M1gXL = M2g(1-X)L => X(M1+M2) = M2 (as FD1 = FD2 - remember nat 5 engineering??). As can be expected, the centre of mass sits on a line between two particles - on the midpoint if they are of equal mass, and tending toward the heavier if not. This is usually given in a system where 0<X<1.

In a multiple particle system of varying mass, the net force on a given particle is the sum of any external force plus all internal forces exerted by other particles within the system. However the summation of internal force is 0, as by Newton's third law each will find its counterpart and cancel out. This describes the trajectory of a centre of mass for a macroscopic body made of multiple particles:


 * rcom = 1/M ΣNundefinedmiri
 * Mvcom = ΣNundefinedmivi
 * Macom = ΣNundefinedmiai = net force = external force

This justification allows scientific use of point masses when describing large systems.

Impulse
The linear momentum of a single particle is a constant of motion, given as p = mv. It changes according to Newton's second law - dp/dt = ma = F. Integration of momentum between times gives pfinal - pinitial = ∫F(t) dt. If the impulse of a force (measured in Ns or kgm/s) is its change in momentum, then J = F(t) dt. This is the impulse-momentum theorem, and must be applied separately for momentum in x,y and z directions.

In the event of a constant force, J = FΔt. In practice this is another lie of omission, but it makes for a good estimate for an average impulse. Impulse can be read as the area under a force-time graph, using an average where force is taken as half of the maximum force.

Example

 * Impulse on a ball impacted by a bat - this example can be taken as 1D, and uses p1 = p2 where p1 = -mv1 and p2 = mv1 in order to take into account the direction change. These are used to find the impulse as change in momentum; from this if the time is known an average force can be calculated, and from that an average acceleration.

Summary
Particles interact in collisions, but retain a constant overall momentum. In a macroscopic object the net force is equal to the external force, as internal forces cancel out. Newton's second law allows Force = dp/dt = Ma = Mdv/dt = Mdr2/dt2. Momentum is conserved in the absence of external forces. Impulse is the change in momentum and given by J = F(t) dt.