This document provides an overview of further mechanics concepts including: 1) Linear momentum and its conservation, where the total momentum in a system before and after collisions is constant. 2) Elastic and inelastic collisions in one dimension, where momentum is conserved in all collisions but energy is only conserved in elastic collisions. 3) Two-dimensional collisions can be analyzed by splitting momentum into its x and y components. 4) Applications including the conservation of momentum in rocket launches where the rocket gains upwards momentum as the exhaust gases gain downwards momentum.

1. PHYSICS
Further Mechanics
Andres Igea
Friday 19th October 2012

2. Further Mechanics
Linear Momentum
Conservation of Linear Momentum
Target check
Elastic and Inelastic Collisions in one
dimension
Target check
Two-dimensional collisions
Applications
Resources

3. Linear Momentum
The linear momentum of a particle of
mass m and velocity v is defined as:
 
p = m⋅v
The linear momentum is a vector
quantity. Its direction is along v.

4. Conservation of Linear Momentum
 
p = ∑ p = constant
or:
 
∑ pi =∑ p f
  
p1,i + p2,i = p1, f + p2, f

5. Target check
A 2kg marble travels to the right at 0.4 m/s, on
a smooth, level surface. It collides head-on with
a 6kg marble moving to the left at 0.2 m/s.
After the collision, the 2 kg marble rebounds at
0.1 m/s.
Task:
Find the velocity of the 6kg marble after the
collision.
Why are the marbles
so heavy ?

6. Elastic and Inelastic Collisions in
one dimension
Momentum is conserved in any collision,
elastic and inelastic.
Mechanical Energy is only conserved in
elastic collisions.
Perfectly inelastic collision: After colliding, particles coalesce
(stick together). There is a loss of energy.
Elastic collision: Particles bounce off each other without loss
of energy.
Inelastic collision: Particles collide with some loss of energy
(deformation), but don’t coalesce.

7. Perfectly inelastic collision of two
particles
 
 pi = p f Notice that p and v
 are   are vector
m1v1i + quantities = (m1 + m2 )v f
m2 v2i
 and, thus have a
direction (+/-).
Ki − Eloss = K f
1 2 1 2 1 2
m1v1i + m2v2i = ( m1 + m2 )v f + Eloss
2 2 2
There is a loss in energy Eloss

8. Elastic collision of two particles
Momentum is conserved
   
m1v1i + m2 v2i = m1v1 f + m2 v2 f
Energy is conserved
1 2 1 2 1 2 1 2
m1v1i + m2 v2i = m1v1 f + m2 v2 f
2 2 2 2

9. Target check
A bullet (m = 0.01kg) is fired into a block (0.1 kg) sitting at the edge of a
table. The block (with the embedded bullet) flies off the table (h = 1.2 m)
and lands on the floor 2 m away from the edge of the table.
a.) What was the speed of the bullet?
b.) What was the energy loss in the bullet-block collision?
vb = ?
h = 1.2 m
x=2m

10. Two-dimensional collisions
Two particles:
Conservation of momentum:
 
pi = p f
   
m1v1i + m2 v2i = m1v1 f + m2 v2 f
Split into components:
p x ,i = p x , f
m1v1ix + 2 v2 ix =m1v1 fx + 2 v2 fx
m m
p y ,i = p y , f
m1v1iy + 2 v2 iy =m1v1 fy + 2 v2 fy
m m

11. Applications
Conservation of momentum:
Rocket being launched into space.
Rocket gains
momentum in
the upwards
direction
The hot gases
gain
momentum in
the
downwards
direction

12. Resources
AQA Mechanics 1 book
EDEXCEL Physics A2 Textbook
www.physicsclassroom.com
www.nasa.gov

13. Thanks a lot for
watching
Andres