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Lecture 18

An Introductory Physics Course with Peter Eyland

Lecture 17 (Momentum conservation)

In this lecture the following are introduced:

• Conservation of momentum

• Inelastic and Elastic Collisions

• Explosions

• Rockets

Newton's first law says:

*Bodies remain in a state of rest, or of constant speed in a straight line, unless compelled to change by a push or a pull*.

This may be re-phrased as follows:

In the absence of external forces on a system the momentum of the system stays unchanged (i.e. momentum is said to be *conserved*).

Collisions

The diagram shows |

Momentum of centre of mass before collision = momentum of centre of mass after collision

**p**_{before} = **p**_{after}

m_{1}·**u**_{1} + m_{2}·**u**_{2} =
m_{1}·**v**_{1} + m_{2}·**v**_{2}

Inelastic and Elastic Collisions

Collisions are characterised by how quickly the objects approach and how quickly they separate. In this lecture, the two extreme cases are considered.

In perfectly inelastic collisions the objects stick together; and to do this,
energy is required to change their shape.
It means that *Kinetic Energy is not conserved* in perfectly inelastic collisions.

In perfectly elastic collisions the objects bounce off and
regain their shape perfectly.
It means that *Kinetic Energy is conserved* in completely elastic collisions.

Example

A police constable of mass 65kg (including a bullet-proof vest) is at rest on an ice rink (no friction) and aiming at an armed offender.
The offender (a lapsed member of the NRA) shoots first, firing right to left.
The constable's vest absorbs the bullet and stops it without any penetration.
The velocity of the bullet is 320 m.s^{-1} on collision and its mass is 50 g.
Find the final velocity of the constable and bullet assuming all motion is in a straight line.

A convention for direction needs to be specified first. Take left to right as positive and right to left as negative.

The negative sign indicates that the constable moves right to left, ie, same direction as the bullet.

Example

A car with mass 1500 kg travelling North at 92 km.hr^{-1} is struck by another car of mass 2000 kg travelling East at 60 km.hr^{-1}.
The cars lock together on collision.
Find the resulting velocity of the combined cars just after collision.

A convention for direction needs to be specified first. Take North as up the screen and East to the right.

From Pythagoras' theorm From the definition of the tangent The angle is (90-49)=41 |

Example

A billiard ball of mass 0.5 kg travelling with speed 8 m.s^{-1} hits the cushion at an angle of incidence of 41.4°
and rebounds without loss of speed so that the angle of reflection equals the angle of incidence. The collision takes 0.2 ms. Find

(a) the change in the momentum of the ball, and

(b) the force of the cushion on the ball.

A convention for direction needs to be specified first. Take North as up the screen and East to the right.

A vector difference is the second vector minus the first,
so take the second vector and add a first vector in the opposite direction.
For the size: The direction is at right angles away from the wall (West). |

Explosions

An explosion is a kind of time-reversed collision.
In an explosion, parts of the initial object exert forces on other parts and so fly apart from each other.
As only internal forces are acting, momentum is conserved and the centre of mass of the system is not affected.

Example

An 80 kg man is standing on a 6.5 kg stationary sled which is on a frictionless ice surface.
The man jumps horizontally off the back of the sled with a velocity of 3.5 m.s^{-1} due South.
Find the sled's velocity immediately after the man jumps off.

A convention for direction needs to be specified first. Take North as up the screen and East to the right.
Also, take South as positive.

The sled travels North, very quickly!

Example

A radioactive nucleus, initially at rest, decays by emitting an electron and a neutrino at right angles to one another.
In one such decay the momentum of the electron is 1.2 x 10^{-22} kg.m.s^{-1}
and the momentum of the neutrino is 6.4 x 10^{-23} kg.m.s^{-1}.
The mass of the residual nucleus is 5.8 x 10^{-26} kg.
Find the velocity of the recoiling nucleus.

Let the neutrino move in the positive x direction, and the electron move in the positive y direction.

Since the initial momentum is zero, the resulting momenta must add up to zero, i.e. they must form a closed polygon.
In this case it will be a right angled triangle.
Multiplying all sides by 10 The nucleus moves in a direction which is 152° anticlockwise from the electron and 118° clockwise from the neutrino. |

Rockets

A rocket works when internal fuel is exploded and propelled backwards.
Since one part of the system (the exploded fuel) acts on another part of the system (the body of the rocket)
there are no external forces acting on that system, and momentum is conserved.

Writing Newton's second law for this system, the force is zero, but the mass is varying, i.e.

Just considering the body of the rocket as the system, the exploded fuel provides an external forward force
(*Thrust = m _{body}·a_{body}*).
This thrust depends on the relative speed of the exploded fuel (i.e. the exhaust gases)
and the amount of fuel per second which is thrown backwards.
If the amount of fuel per second remains constant then acceleration will increase with the dcreasing mass of the body of the rocket.
As the thrust does not depend on any outside influence this will work in the vacuum of space.

*Summarising:*

In the absence of external forces on a system the momentum of the system is conserved.

Momentum is conserved in collisions, explosions and with rockets.

Kinetic Energy is not conserved in perfectly inelastic collisions.

Kinetic Energy is conserved in perfectly elastic collisions.

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