Module : Conservation of Momentum

Kreshnik Angoni and Kevin Lenton
Worksheet.docx
Worksheet.pdf

Other Resources:

• Haliday & Resnick, Fundamentals of Physics 9.4-11
• Openstax
• Printable version

How to Save the Earth from Catastrophe

http://static.ddmcdn.com/gif/asteroid-hits-earth-2.jpg
Asteroids hit the earth every year, and it is just a matter of time before ‘the big one’ has a path close enough to the earth to cause concern.
How would you save Earth from such an asteroid?
The most obvious choice is to send a nuclear bomb up to the asteroid and explode it (as seen in the movie Armageddon). You hope that this big force applied for a short amount of time will be enough to change the orbit of the asteroid such that it misses the earth. This probably will not work very well because the bomb will probably just split the asteroid up into smaller fragments – all still heading for Earth!
Experts recommend, rather, shining a powerful laser onto the asteroid. The force from the laser onto the asteroid is small, but by applying it for a long time you can get the same effect of changing the path of the asteroid.
The point is that applying a small force for a long time can have the same effect as applying a big force for a short time. The Force times Time is the same in the two cases.
This value, the product of Force and Time is called the Impulse J. More correctly, it is the integral of force times time. ${\vec {J}}=\int {\vec {F}}dt$
Note that the direction of the Impulse ${\vec {J}}$ is in the same direction as the force ${\vec {F}}$.
If the force is constant, or assumed to be constant then:
${\vec {J}}={\vec {F}}\times t$
(Also just a quick note to point out that Work is the product of Force and Displacement, whereas Impulse is the product of Force and Time.)

Impulse is equal to the Change in Momentum, momentum is mass times velocity

Starting with this definition we can use Newton’s Second Law to derive an expression for momentum:
${\vec {F_{net}}}=m{\vec {a}}$
Multiply both sides by dt
${\vec {F_{net}}}dt=m{\vec {a}}dt$
Take the integral of both sides, from the initial to the final condition
$\int {\vec {F_{net}}}dt=\int _{initial}^{final}m{\vec {a}}dt$
Rewrite using the definition of J:
${\vec {J_{net}}}=\int _{i}^{f}m{\vec {a}}dt$
Rewrite the acceleration in terms of velocity:
${\vec {J_{net}}}=\int _{i}^{f}m{\frac {d{\vec {v}}}{dt}}dt=\int _{i}^{f}md{\vec {v}}$
therefore
${\vec {J_{net}}}=m{\vec {v_{f}}}-m{\vec {v_{i}}}$
This value $m{\vec {v}}$ is called the Momentum and is given the symbol ${\vec {p}}=m{\vec {v}}$
Therefore:
${\vec {J_{net}}}={\vec {p_{f}}}-{\vec {p_{i}}}$
or
${\vec {J_{net}}}=\Delta {\vec {p}}$

That is to say Impulse is equal to the change in Momentum.

The units of impulse and momentum are either: $Ns$ (from the definition of Impulse), or $kg{\frac {m}{s}}$ from the definition of momentum.

The Conservation of Momentum depends on the System

Consider the universe as a closed, isolated system. Any force in the universe is internal to the universe and has an equal opposite reaction on some other object (Newton's Third Law). The force imparts Impulse to the object that it is acting on, but the reaction to this force (which is equal and opposite)produces an equal and opposite Impulse. The vector sum of these two impulses sum to zero. That is to say for every impulse a force gives to the universe, there is an equal and opposite impulse somewhere else. In other words all you are doing is transferring impulse from one object in one part of the universe to another object in another part of the universe.
The net impulse is zero, and therefore the change in momentum is zero. ${\vec {J_{net}}}={\vec {p_{f}}}-{\vec {p_{i}}}=0$
or
${\vec {p_{f}}}={\vec {p_{i}}}$
That is to say Momentum is conserved for a closed system with no external Impulses.

For a systme with external Impulses, the general equation, for a system smaller than the universe is
${\vec {J_{net}}}={\vec {p_{f}}}-{\vec {p_{i}}}\neq 0$
That is to say the change in momentum is equal to any impulse coming in, or leaving the system.

Momentum: What is it good for?

Momentum is a very useful way of describing interactions, particularly collisions between objects. As we have seen, Conservation of Momentum is simply another way of describing Newton's Second and Third Law.
Normally the forces between objects colliding, for example, are quite complicated, being dynamic and non-linear. You would need sophisticated computer models to predict how objects would move during the collision, if you just looked at the forces.
However, we know that the momentum is conserved in an isolated system, so that the momentum before and after the collision is conserved. This means that you can use momentum to calculate the motion of the system components after the collision, without having to know the exact forces involved. Very useful!
Ultimately however, momentum is simply another way of describing the effect of force.

Misconception #1: "Momentum is Inertia"

Momentum is not inertia.
The concept of inertia is really the concept of mass. Mass describes how an object accelerates given a net force. Therefore inertia describes how easy it is to change the velocity of an object.
Momentum is the mass times the velocity, something quite different. Momentum is really only useful when looking at the change in momentum of a system. But, if it helps you, the momentum of an object can be thought of as a measure of how much impulse (a force applied for a length of time)is required to bring the system to rest.

Misconception #2: "Momentum is always conserved"

Momentum is only conserved when:
${\vec {J_{net}}}=0$
That is to say the net force on the system is zero, and therefore no net impulses are acting from outside the system, the system is closed.
This depends on the definition of your system!
If the universe is the system, then yes, momentum is always conserved. Anything less, and you have to determine if the system is closed or not. If there are external forces to the system, adding impulse, you will have to use:
${\vec {J_{net}}}={\vec {p_{f}}}-{\vec {p_{i}}}\neq 0$

Types of Collisions: Elastic and Inelastic

The principle of linear momentum conservation is very general, it can be used to apply to any sort of force interaction. It applies in all fields of physics, mechanical collisions, explosions, reactive motion, light emission/absorption, nuclear decay and nuclear reactions. So, it is important to clarify some basic issues related to term “collisions”.
Collisions can be classified by the energy transfer into or out of the system.
A collision where no kinetic energy is lost during the collision is called a completely elastic collision. An example of an elastic collision would be two electrons colliding. The electrons do not actually touch and no energy is lost due to e.g. friction etc.
A collision where maximum kinetic energy is lost during the collision is called a completely inelastic collision (Note: maximum is not the same as all). An example of a completely inelastic collision is two cars colliding and sticking together. The cars lose energy due to the crumpling of the car body, sound etc. Any collision when the objects stick together is completely inelastic.
In reality, most collisions are somewhere in between: some energy is lost, but the objects do not stick together.
You can know whether a collision is elastic or inelastic by comparing the total kinetic energy of the system before and after the collision.

Examples

Example Impulse and Momentum
An 800g baseball travelling with a velocity equal to 5m/s hits the ground at an angle of 40 degrees to the horizontal. The ball then rebounds at an angle of 40 degrees with the same speed. The ball is in contact with the ground for 0.006s.
Find the impulse of the ground on the ball.
Find the average force of the ground on the ball.

Example: 1D Conservation of Momentum: Elastic Collision
Two identical 1.50-kg masses are pressed against oppo- site ends of a light spring of force constant 1.75 N/cm, compress- ing the spring by 20.0 cm from its normal length. Find the speed of each mass when it has moved free of the spring on a frictionless horizontal table.

Example 2D Momentum: Inelastic Collision
At the intersection of Ste.Croix Avenue and Cote-Vertu Blvd, a yellow subcompact car with mass 950 kg traveling east on Cote-Vertu collides with a red pickup truck with mass 1900 kg that is traveling north on Ste. Croix and has run a red light. The two vehicles stick together as a result of the collision, and the wreckage slides at 16.0 m/s in the direction 24.0° east of north. Calculate the speed of each vehicle before the collision. The collision occurs during a heavy rainstorm; you can ignore friction forces between the vehicles and the wet road.