Momentum is another vector measurement. Momentum is in the same direction as velocity. We calculate momentum by multiplying the mass of the object by the velocity of the object. It is an indication of how hard it would be to stop the object. If you were running, you might have a mass of 50 kilograms and a velocity of 10 meters per second west (really fast). Your momentum would be 500 kg-m/sec west. Remember Newtons first law? It said that any object moving will continue moving unless it is interfered with. That idea applies to momentum as well. The momentum of an object will never change if it is left alone. If the 'm' value and the 'v' value remain the same, the momentum value will be constant.

The momentum of an object, or set of objects (system), remains the same if it is left alone. Within such a system, momentum is said to be conserved. Here's the momentum idea in simpler terms. When you throw a ball at someone and it hits him hard, it hurts because it was difficult to stop (had momentum).Think about it. If you throw a small ball and a large ball at the same speeds, the large ball will hit a person with a greater momentum, be harder to stop, and hurt more. When the mass is greater (at the same speeds), the momentum is greater.

A bullet is an example of an object with a very small mass that has a lot of momentum because it is moving very quickly. Bullets are therefore difficult to stop; it's a good idea not to try!

If we have a system that consists of two objects we can construct the momentum of the system:

p(system) = p(obj 1) + p(obj 2)

where it is emphasised that the addition is done as vectors. Generalising to
many diﬀerent objects is simply a case of adding more momenta.
The reasons that momentum is so important are two-fold:

- Momentum is a
. More explicitly, to know the momentum of an object we only need to know its mass and velocity*state function**at that instant.* - The momentum of a closed momentum system is conserved.
A closed momentum system means a system that is not being “pushed” or
“pulled” overall by outside inﬂuences.

This is diﬀerent from a closed energy system(where energy remains constant). It is possible to have an open energy system and a closed momentum system (e.g. an object being heated while at rest), or a closed energy system and an open momentum system. As an exercise see if you can come up with an example of an closed momentum system that is an open energy system (they do exist). The way we determine if a momentum system is open or closed is by studying the pushes and pulls – known in physics as forces – which we will do next. For the time being, we will use our intuition for pushes and pulls.

The above properties of momentum make it very similar to energy. The
basic approach to solving problems using momentum conservation will be
the same as the approach to energy conservation. We pick an initial and
ﬁnal time, and ask what the momentum has to be in order to be conserved.
Because momentum is a state function, we don’t have to worry about the
messy details between those two times provided that the system was closed
throughout.

Thelaw of conservation of linear momentumstates that if no external forces act on the system of two colliding objects, then the vector sum of the linear momentum of each body remains constant and is not affected by their mutual interaction.

Alternatively, it states that if net external force acting on a system is zero, the total momentum of the system remains constant.

Proof:

Let us consider a particle of mass ‘m’ and acceleration ‘a’. Then, from second law of motion,

If no external force acts on the body then, F=0,

Therefore, ‘P’ is constant or conserved.

(*Note: If the derivative of any quantity is zero, it must be a constant quantity.*)

**Deduction of Law of Conservation of linear momentum for two colliding bodies**

Let us consider two bodies of masses m1and m2 moving in straight line in the same direction with initial velocities u1and u2. They collide for a short time ∆t. After collision, they move with velocities v1and v2.From second law of motion,

Force applied by A on B = Rate of change of momentum of B

FAB = (m2*v2 - m2*u2)/∆t

Similarly, Force applied by B on A = Rate of change of momentum of A

FBA = (m1*v1 - m1u1)/∆t

From Newton’s third law of motion,

FAB = -(FBA)

or, (m2*v2 - m2*u2)/∆t = -(m1*v1 - m1u1)/∆t

or, (m2*v2 - m2*u2) = -(m1*v1 - m1u1)

or, (m1*u1 + m2*u2) = (m1*v1 + m2v2)

This means the total momentum before collision is equal to total momentum after collision. This proves the principle of co conservation of linear momentum.

Impulse is simply another name for ∆p

Any object with momentum is going to be hard to stop. To stop such an object, it is necessary to apply a force *against* its motion for a given period of time. The more momentum that an object has, the harder that it is to stop. Thus, it would require a greater amount of force or a longer amount of time or both to bring such an object to a halt. As the force acts upon the object for a given amount of time, the object's velocity is changed; and hence, the object's momentum is changed.

A force acting for a given amount of time will change an object's momentum. Put another way, an unbalanced force always accelerates an object - either speeding it up or slowing it down. If the force acts opposite the object's motion, it slows the object down. If a force acts in the same direction as the object's motion, then the force speeds the object up. Either way, a force will change the velocity of an object. And if the velocity of the object is changed, then the momentum of the object is changed.

If both sides of the above equation are multiplied by the quantity t, a new equation results.

This equation represents one of two primary principles to be used in the analysis of collisions during this unit. To truly understand the equation, it is important to understand its meaning in words. In words, it could be said that the force times the time equals the mass times the change in velocity. In physics, the quantity** Force • time** is known as

*impulse*

change in momentum. The equation really says that the

Impulse = Change in momentum

One focus of this unit is to understand the physics of collisions. The physics of collisions are governed by the laws of momentum; and the first law that we discuss in this unit is expressed in the above equation. The equation is known as the impulse-momentum change equation

. The law can be expressed this way:In a collision, an object experiences a force for a specific amount of time that results in a change in momentum. The result of the force acting for the given amount of time is that the object's mass either speeds up or slows down (or changes direction). The impulse experienced by the object equals the change in momentum of the object. In equation form, F • t = m • Δ v.

A collision is an event where momentum or kinetic energy is transferred from one object to another. Momentum (p) is the product of mass and velocity (p = mv). A large truck massing 10,000 kg and moving at 2 meters/sec has the same momentum as a 1,000 kg compact car moving at 20 meters/sec; they both have p = 20,000 kg m/sec. The other quantity that can be transferred in a collision is kinetic energy. Kinetic energy is the energy of motion; it is defined as K = (1/2) m v^2. The relationship between kinetic energy and mass is linear, which means that a vehicle massing twice as much has twice as much kinetic energy. The relationship between kinetic energy and velocity is exponential, which means that as you increase your speed, kinetic energy increases dramatically.

There are two general types of collisions in physics: elastic and inelastic. An inelastic collisions occurs when two objects collide and do not bounce away from each other.

Momentum is conserved, because the total momentum of both objects before and after the collision is the same. However, kinetic energy is not conserved. Some of the kinetic energy is converted into sound, heat, and deformation of the objects. A high speed car collision is an inelastic collision. In the above example, if you calculated the momentum of the cars before the collision and added it together, it would be equal to the momentum after the collision when the two cars are stuck together. However, if you calculated the kinetic energy before and after the collision, you would find some of it had been converted to other forms of energy.

An elastic collision occurs when the two objects "bounce" apart when they collide. Two rubber balls are a good example.

In an elastic collision, both momentum and kinetic energy are conserved. Almost no energy is lost to sound, heat, or deformation. The first rubber ball deforms, but then quickly bounces back to its former shape, and transfers almost all the kinetic energy to the second ball.

A car's bumper works by using this principle to prevent damage. In a low speed collision, the kinetic energy is small enough that the bumper can deform and then bounce back, transferring all the energy directly back into motion. Almost no energy is converted into heat, noise, or damage to the body of the car, as it would in an inelastic collision.

However, car bumpers are often made to collapse if the speed is high enough, and not use the benefits of an elastic collision. The rational is that if you are going to collide with something at a high speed, it is better to allow the kinetic energy to crumple the bumper in an inelastic collision than let the bumper shake you around as your car bounces in an elastic collision. Making their bumpers this way benefits the car companies: they get to sell a new bumper, and you can't sue them for whiplash.

- Momentum is a vector, and points in the direction of motion.
- Momentum of a system is conserved if nothing is pushing or pulling it from outside the system.
- An elastic collision is a collision that conserves kinetic energy.
- An inelastic collision is a collision that does not conserve kinetic energy.
- All collisions (in a closed momentum system) conserve momentum.
- Impulse is another name for “change in momentum”, and is sometimes denoted
.*J*