Linear DynamicsMomentum and motion of systems.

Collisions.

o Linear Dynamics Nature of Momentum (& Energy), origins Linear Momentum Conservation Momentum for a system of particles

o Collision and Impulse Elastic Collisions Inelastic Collisions

o Aim of the lecture Concepts in Collisions

Dependence on relative motion Dependence on masses Dependence on speed

Newton’s Second Law Conservation of momentum

o  Main learning outcomes familiarity with

Two body Collisions (particles!) Use of energy and momentum conservation Centre of Mass for a system

Lecture 7

Linear Momentum

o Conserved Quantities If there is a symmetry in nature, then There will be a conserved quantity associated with it.

• (the maths to prove this is beyond the scope of the course)o Examples:

Physics today is the same as physics tomorrow, TIME symmetry The conserved quantity is Energy

Physics on this side of the room is the same as on the other side Linear Translational symmetry The conserved quantity is called Momentum

o What is momentum? For a mass m moving at speed v, the momentum, p is

p = mv

o In fact momentum is a vector quantity, so we should write

p = mv, where v and p are vectors.

vm

Centre of Mass Frame of Reference

o The momentum will depend on the mass and the speed

But speed depends on where you measure from. The momentum is defined relative to a reference frame, here it is the woman watching the ball who is ‘stationary’

p=mvv

Centre of Mass Frame of Reference

o The man sees the ball as stationary because he is moving with it the mass is the same but the speed as measured by the man is zero

The momentum is zero in this ‘reference frame’It is called the ‘zero momentum frame’ or ‘the centre of mass’

p=mv

p = m0 = 0

o For several objects considered together ‘a system’o The total momentum is the sum of the individual momenta

P = pi

Where: pi is the momentum of the ith particle means ‘sum of’ P is the total momentum

P = p1 + p2 = 2mv

The total momentum of the two balls isthe sum of their individual momenta

v

v

o For several objects considered together ‘a system’o In this case the balls move in opposite directions, so the total momentum is zero

P = pi

v

P = p1 + p2 = mv – mv = 0

The total momentum of the two balls isthe sum of their individual momenta

v

o Momentum is conserved Must stay in one frame of reference In collisions Always in fact

o Energy is also conservedo Together energy and momentum conservation

Solve collision problems

What happens after collision?

Before collision total momentum P = mv-mv=0

v v

Total P = 0 so the balls could just stop

After Collision

Total P = 0 so balls could bounce

Total P = mv + 0 = mv this is not zero and so CANNOT happen

What happens depends on energy conservationas well as momentum conservation

If all the energy remains as kinetic energy, theThe buses will simply bounce back the way they came

?

At the same speedthey came in

v v

If all the energy is absorbed in a crash, in sound/deformation

?v v

The buses will simply crash and stop

?v v

P = 0, so after collision p1 = -p2

If energy remains as kinetic energy E1+E2= const

E1 =E2 = mv2/2 = p2/2m

so for both energy and momentum to be conservedthe buses must simply bounce back at same speed

?v v

P = 0, so after collision p1 = -p2

If energy is converted into sound/deformation in collision

Kinetic energy: E1 = E2 = 0 after collision = mv2/2 = p2/2m

so for both energy and momentum to be conservedthe buses must stop

o More generally Consider two unequal masses and speeds In elastic collision energy and momentum are conserve

v1i

What happens after collision?

Before collision total momentum P = 2mv + 0 = 2mv total energy E = 2mv2/2 + 0 = 2mv2/2

2mm v2i=0

v1f

Before collision total momentum P = mv1i + mv2i

total energy E = m(v1i)2/2 + m(v2i)2/2

2mm v2f

After the collision both energy and momentumremain the same

With some algebra: v1f = (m1-m2)v1i/(m1+m2) = v1i/3 v2f = 2m1v1i/(m1+m2) = 4v1i/3

Newton’s Cradle - explain