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  Impulse and Linear Momentum, Physics tutorial

Impulse:

These concepts are just the outgrowth of Newton's second law. Newton's second law (Fnet = m.a) expresses that acceleration of the object is directly proportional to net force applying on object and inversely proportional to mass of object. When joined with definition of acceleration (a = change in velocity / time), the given equalities result.

F = m.a

or

F = m . Δv / t

If both sides of above equation are multiplied by quantity t a new equation results:

F.t = m.Δv

This equation signifies one of two main principles to be used in analysis of collisions during this unit. To truly know the equation, it is significant to know its meaning in words. In words, it could be stated that force times time equals the mass times the change in velocity. The quantity Force X time is called as impulse. And as quantity m.v is momentum, the quantity m. Δv should be change in momentum. Equation actually says that

Impulse = Change in momentum

Momentum:

It is a quantity which explains the object's resistance to stopping (a type of moving inertia). It is represented by symbol p (boldface). It is a product of the object's mass and velocity.

p = m v

It is the vector quantity (as velocity is vector and mass is a scalar). It uses SI unit kilogram meter per second [kg m/s].

Motion with Variable Mass:

If the mass of a system varies with time, we can express Newton's second law of motion as

F = dp/dt =d(mv)/dt = md(v/dt)+v(dm/dt)

Under special case when v is constant, Equation becomes

F = v(dm/dt)

Motion of a Rocket:

Rockets range in size from fireworks so small that ordinary people utilize them to immense Saturn Vs which once propelled huge payloads toward Moon. Propulsion of all rockets, jet engines, deflating balloons, and even squids and octopuses are described by same physical principle: Newton's third law of motion. Matter is vigorously ejected from the system, generating an equal and opposite reaction on what remains. Another common instance is recoil of the gun. Gun applies the force on the bullet to accelerate it and as a result experiences the equal and opposite force, causing gun's recoil or kick.

The rocket has a mass m and a velocity v relative to Earth, and therefore the momentum mv. In part, a time Δt has elapsed in which rocket has ejected the mass Δm of hot gas at velocity ve relative to rocket. Remainder of mass (m-m) now has the greater velocity (v+ Δv). The momentum of complete system (rocket plus expelled gas) has really decreased as force of gravity has acted for the time Δt, producing the negative impulse Δp=-mgΔt. So center of mass of system is in free fall but, by quickly expelling mass, part of system can accelerate upward. It is a commonly held delusion that rocket exhaust pushes on ground. If we consider thrust; i.e., force applied on rocket by exhaust gases, then the rocket's thrust is greater in outer space than in atmosphere or on launch pad. Actually, gases are easier to expel in the vacuum.

By computing change in momentum for complete system over Δt, and equating the change to impulse, the given expression can be illustrated to be a good approximation for acceleration of rocket.

α = (ve/m)(Δm/Δt) - g

where α is acceleration of rocket, ve is escape velocity, m is mass of rocket, Δm is mass of ejected gas, and Δt is the time in which gas is ejected.

Linear Momentum:

Linear momentum is the vector quantity stated as product of the object's mass, m, and its velocity, v. Linear momentum is signified by letter p and is known as momentum for short:

p= mv

A body's momentum is always in same direction as its velocity vector. Units of momentum are kg.m/s.

Newton's Second Law as F = ma. Though, as acceleration can be stated as Δv/Δt, express Newton's Second Law as F = mΔv/Δt. Substituting p for mv, we find an expression of Newton's Second Law in terms of momentum:

F = Δp/Δt

In fact, this is form in which Newton first stated his Second Law. It is more flexible than F = ma as it can be utilized to examine systems where not just the velocity, but also mass of the body changes, as in case of the rocket burning fuel.

Conservation of Linear Momentum:

In the special case when the net external force Fe is zero,

dp/dt = 0

so that p = p1 + p2 = a constant vector. This is principle of conservation of linear momentum for a two particle system. It is evenly valid for the system of any number of particles. Its formal proof for the many-particle system will be given later. It expresses that:

If the net external force acting on the system is zero, then total linear momentum is conserved.

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