Collisions © D Hoult 2010

Elastic Collisions © D Hoult 2010

Elastic Collisions 1 dimensional collision © D Hoult 2010

Elastic Collisions 1 dimensional collision: bodies of equal mass © D Hoult 2010

Elastic Collisions 1 dimensional collision: bodies of equal mass (one body initially stationary) © D Hoult 2010

Elastic Collisions 1 dimensional collision: bodies of equal mass (one body initially stationary) © D Hoult 2010

Before collision, the total momentum is equal to the momentum of body A AB uAuA © D Hoult 2010

After collision, the total momentum is equal to the momentum of body B AB vBvB © D Hoult 2010

The principle of conservation of momentum states that the total momentum after collision equal to the total momentum before collision (assuming no external forces acting on the bodies) © D Hoult 2010

The principle of conservation of momentum states that the total momentum after collision equal to the total momentum before collision (assuming no external forces acting on the bodies) m A u A = m B v B © D Hoult 2010

The principle of conservation of momentum states that the total momentum after collision equal to the total momentum before collision (assuming no external forces acting on the bodies) m A u A = m B v B so, if the masses are equal the velocity of B after © D Hoult 2010

The principle of conservation of momentum states that the total momentum after collision equal to the total momentum before collision (assuming no external forces acting on the bodies) m A u A = m B v B so, if the masses are equal the velocity of B after is equal to the velocity of A before © D Hoult 2010

Bodies of different mass © D Hoult 2010

AB

AB uAuA

AB uAuA Before the collision, the total momentum is equal to the momentum of body A © D Hoult 2010

AB vAvA vBvB

AB vAvA vBvB After the collision, the total momentum is the sum of the momenta of body A and body B © D Hoult 2010

AB vAvA vBvB If we want to calculate the velocities, v A and v B we will use the © D Hoult 2010

AB vAvA vBvB If we want to calculate the velocities, v A and v B we will use the principle of conservation of momentum © D Hoult 2010

The principle of conservation of momentum can be stated here as © D Hoult 2010

m A u A = m A v A + m B v B The principle of conservation of momentum can be stated here as © D Hoult 2010

m A u A = m A v A + m B v B If the collision is elastic then The principle of conservation of momentum can be stated here as © D Hoult 2010

m A u A = m A v A + m B v B If the collision is elastic then kinetic energy is also conserved The principle of conservation of momentum can be stated here as © D Hoult 2010

m A u A = m A v A + m B v B If the collision is elastic then kinetic energy is also conserved ½ m A u A 2 = ½ m A v A 2 + ½ m B v B 2 The principle of conservation of momentum can be stated here as © D Hoult 2010

m A u A = m A v A + m B v B If the collision is elastic then kinetic energy is also conserved m A u A 2 = m A v A 2 + m B v B 2 ½ m A u A 2 = ½ m A v A 2 + ½ m B v B 2 The principle of conservation of momentum can be stated here as © D Hoult 2010

From these two equations, v A and v B can be found m A u A = m A v A + m B v B m A u A 2 = m A v A 2 + m B v B 2 © D Hoult 2010

From these two equations, v A and v B can be found m A u A = m A v A + m B v B m A u A 2 = m A v A 2 + m B v B 2 BUT © D Hoult 2010

It can be shown* that for an elastic collision, the velocity of body A relative to body B before the collision is equal to the velocity of body B relative to body A after the collision © D Hoult 2010

It can be shown* that for an elastic collision, the velocity of body A relative to body B before the collision is equal to the velocity of body B relative to body A after the collision * a very useful phrase ! © D Hoult 2010

It can be shown* that for an elastic collision, the velocity of body A relative to body B before the collision is equal to the velocity of body B relative to body A after the collision In this case, the velocity of A relative to B, before the collision is equal to uAuA © D Hoult 2010

It can be shown* that for an elastic collision, the velocity of body A relative to body B before the collision is equal to the velocity of body B relative to body A after the collision In this case, the velocity of A relative to B, before the collision is equal to u A uAuA © D Hoult 2010

It can be shown* that for an elastic collision, the velocity of body A relative to body B before the collision is equal to the velocity of body B relative to body A after the collision and the velocity of B relative to A after the collision is equal to vAvA vBvB © D Hoult 2010

It can be shown* that for an elastic collision, the velocity of body A relative to body B before the collision is equal to the velocity of body B relative to body A after the collision and the velocity of B relative to A after the collision is equal to v B – v A vAvA vBvB © D Hoult 2010

It can be shown* that for an elastic collision, the velocity of body A relative to body B before the collision is equal to the velocity of body B relative to body A after the collision for proof click hereclick here and the velocity of B relative to A after the collision is equal to v B – v A vAvA vBvB © D Hoult 2010

We therefore have two easier equations to “play with” to find the velocities of the bodies after the collision equation 1 © D Hoult 2010

We therefore have two easier equations to “play with” to find the velocities of the bodies after the collision equation 1 equation 2 m A u A = m A v A + m B v B © D Hoult 2010

We therefore have two easier equations to “play with” to find the velocities of the bodies after the collision equation 1 equation 2 m A u A = m A v A + m B v B u A = v B – v A © D Hoult 2010

AB uAuA

Before the collision, the total momentum is equal to the momentum of body A AB uAuA © D Hoult 2010

After the collision, the total momentum is the sum of the momenta of body A and body B AB vAvA vBvB © D Hoult 2010

Using the principle of conservation of momentum © D Hoult 2010

Using the principle of conservation of momentum m A u A = m A v A + m B v B © D Hoult 2010

Using the principle of conservation of momentum m A u A = m A v A + m B v B AB vAvA vBvB © D Hoult 2010

Using the principle of conservation of momentum m A u A = m A v A + m B v B AB vAvA vBvB © D Hoult 2010

Using the principle of conservation of momentum m A u A = m A v A + m B v B AB vAvA vBvB One of the momenta after collision will be a negative quantity © D Hoult 2010

2 dimensional collision © D Hoult 2010

2 dimensional collision © D Hoult 2010

A B

A B Before the collision, the total momentum is equal to the momentum of body A © D Hoult 2010

After the collision, the total momentum is equal to the sum of the momenta of both bodies © D Hoult 2010

Now the sum must be a vector sum © D Hoult 2010

mAvAmAvA

mBvBmBvB mAvAmAvA

mBvBmBvB mAvAmAvA

mBvBmBvB mAvAmAvA

mBvBmBvB mAvAmAvA

p mBvBmBvB mAvAmAvA

mBvBmBvB mAvAmAvA p mAuAmAuA

mBvBmBvB mAvAmAvA p mAuAmAuA

mBvBmBvB mAvAmAvA p mAuAmAuA

p =mAuAmAuA mBvBmBvB mAvAmAvA p mAuAmAuA © D Hoult 2010

2 dimensional collision: Example Body A has initial speed u A = 50 ms -1 © D Hoult 2010

2 dimensional collision: Example Body A has initial speed u A = 50 ms -1 Body B is initially stationary © D Hoult 2010

2 dimensional collision: Example Body A has initial speed u A = 50 ms -1 Body B is initially stationary Mass of A = mass of B = 2 kg © D Hoult 2010

2 dimensional collision: Example Body A has initial speed u A = 50 ms -1 Body B is initially stationary Mass of A = mass of B = 2 kg After the collision, body A is found to be moving at speed v A = 25 ms -1 in a direction at 60° to its original direction of motion © D Hoult 2010

2 dimensional collision: Example Body A has initial speed u A = 50 ms -1 Body B is initially stationary Mass of A = mass of B = 2 kg After the collision, body A is found to be moving at speed v A = 25 ms -1 in a direction at 60° to its original direction of motion Find the kinetic energy possessed by body B after the collision © D Hoult 2010