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Momentum

Momentum can be defined as “mass in motion.” All objects have mass therefore if  an object is moving, then it has momentum - it has its mass in motion. Momentum depends upon the variables mass and velocity. We can change momentum by changing its mass or changing its velocity, a mass unit is multiplied by a velocity unit to provide a momentum unit. For example, a train moving at 120 km/h, a baseball flying through the air, a heavy truck moving, a bullet fired from a gun, when you throw a ball at someone and it hits him hard. It is an indication of how hard it would be to stop the object. 

As we know all objects have some amount of mass, so when they move or in motion condition then these possess momentum. Momentum has both magnitude and direction as it is a vector quantity. The momentum of a particle is a measure of the time required for a constant force to bring it to rest. The momentum of any object that is at rest is 0. If the mass is kept constant, then the momentum of an object is directly proportional to its velocity. In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object. Where momentum is denoted as “p”, mass is denoted with “m” and velocity is denoted with “v” 

Momentum = mass $\times$ velocity or P = m $\times$ v 

Let us further discuss all about Momentum, its representation, variable factors, and its types.

 

What is Momentum?

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Momentum is a vector because velocity is a vector and mass a scalar. Momentum is defined as the product of mass and velocity. For a particle of mass m and velocity v, the momentum p is mv.

These are certain points which describe the basics of momentum: 
  • Momentum is the product of mass and velocity. 
  • It is a vector quantity and has got the same direction as that of Velocity.
  • The greater the mass or weight the greater is the Momentum. 
  • It is an indication of how difficult it would be to stop a moving object.
  • Momentum is a quantity that has a magnitude, direction and a size. 
  • If the mass is kept constant, then the momentum of an object is directly proportional to its velocity. 

Let us quickly recall the Newton's Law of Motion: 

"The rate of change of momentum of a body is proportional to the net force applied to that body and in the direction of the force." All moving bodies will continue to be in the state of rest or motion unless interfered by some external force. The second law enables one to calculate the magnitude of the resultant force either by measuring the rate of change in momentum of the object to which the force is applied or by measuring the mass and acceleration of the object. The same law applies to momentum as well. That is, if the mass and velocity of an object remain the same then the momentum of the object remains constant, i.e. mv = Constant.

Mass and Velocity
Figure: Depicting the relation of momentum with mass and velocity

Momentum Definition

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The momentum of an object is equal to the mass of the object times the velocity of the object. Where momentum is denoted as “p”, mass is denoted with “m” and velocity is denoted with “v”.
Momentum = mass $\times$ velocity
p = m $\times$ v

S.I Unit of momentum = kgms-1

It is important to note that for a body in motion, the momentum is zero as the velocity is zero.
When the object moves then it gains momentum as the velocity increases. Hence greater the velocity greater is the momentum.

According to the law of conservation of momentum the momentum of the closed system of objects remains conserved given the fact no external force acts on it. 

Or 

If no external force acts on a system in a particular direction then the total momentum of the system in the direction remains unchanged.

For Example:
It is evident from the fact that it is difficult to stop a truck than a bicycle because of the momentum difference. It can also be visualized from the fact that a truck involved in an accident will do more harm than any bicycle.

The quantity of motion of the body or an object is measured by momentum. It can also be called as “The impulse of the moving object”. It should be noted that momentum is a conserved quantity.

Momentum Equation

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According to newton's second law, the rate of change of momentum always acts in the direction of the resultant force acting on the particle and is directly proportional to this force.
Hence sigma F = $\frac{d p}{d t}$
            = $\left \{ \frac{(m)dv}{dt} \right \}+\left \{\frac{(v)dm}{dt} \right \}$
[ As the mass is constant and does not change hence $\frac{(v)dm}{dt}$ = 0 ]
Hence F = m a
Here F is the resultant or net force
The basic formula for momentum of the moving body is:
P = mv ....................(1)
Where,
p = momentum of the body
m = mass
v = velocity of the moving body

Units:
The unit of mass in S.I. system is Kg and
The unit of velocity in S.I. system is m/s
so, the unit of momentum is kgm/s
Also, if velocity is expressed as change of velocity then the formula becomes:
P = m(v1 - v2)...................(2)
Where,
v1 is the final velocity and
v2 is the initial velocity

For a system of particles the formula changes to:
P = $\sigma$ m v
= m1 v1 + m2 v2 + m3 v3 + m4 v4 + . . . . . . . . . . . . .+ mi vi ................(3)
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Units of Momentum

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According to the formula of momentum P = m v, the SI unit is Kgm/s. Also expressed in N-s or Newton second.

Linear Momentum

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Let us suppose the file cabinet is in the middle of the room, a room with a smooth floor, we give it a push in order to move it close to the wall and before we realize, it slams into the wall. It is difficult to stop because it has linear momentum.       The measure of an object’s or body’s translation motion is called Linear Momentum.
  • Linear momentum is a vector quantity.
  • The direction of linear momentum is in the direction of velocity of the object.
  • The formula for linear momentum remains the same, p = m v.
But here we also consider it in different reference frames or different axes, viz. Px, Py,Pz etc.
Here,
Px = m Vx
Py = m Vy
Pz = m Vz
Where Vx, Vy, Vz are the velocities in x, y and z directions respectively.
 
Linear momentum is dependent on the frame of reference:
It is important to note that the an object can have momentum for one frame of reference but the same object if kept in another reference frame can have zero momentum.       
Lets us take an example:
An airplane has a velocity of 20 km/s and mass of 100 kg, then the momentum taking the earth as frame of reference is 20 $\times$ 100 = 2000 Kg Km/s. But to the pilot in the cockpit it has a velocity and momentum of zero.

Linear momentum of a system of particles:
for a system of particles with a mass of m1, m2, m3 and so on and a velocity v1, v2, v3 and so on the linear momentum can be expressed as:
P = $\sigma$ m v
   = m1 v1 + m2 v2 + m3 v3 + m4 v4 + . . . . . . . . . . . . .+  mi vi  .................(a)

Change in Momentum

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We know that,
P = m v
If we consider the velocity to be the initial and final velocity represented by u and v then momentum change or
The change in momentum would be expressed as: 
P = m (v – u)

P = mv – mu
$\Delta P$ = m $\Delta$ vHence, the change in momentum is equal to mass times the change in velocity.
Here mv is the Final Momentum and
        mu
is the Initial Momentum
The change in momentum is also called as Impulse.

Relativistic Momentum

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The momentum defined and used in relativistic mechanics is called Relativistic Momentum. In relativistic mechanics the momentum is defined as follows:
P = y mo v . . . . . . . . . . . . . . . . . . . (1)

Here y is the Lorentz factor.

It is equal to,
y = $\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$
Here,
v is the object's speed
c is the light's speed
mo is the invariant mass
Also the inverse relation can be expressed as:
v = $\frac{c^{2}P}{\sqrt{(pc)^{2} + (m_{0}c^{2})^{2}}}$
Where P = $\sqrt{(P^{2}x + P^{2}y + P^{2}z)}$

The total energy E of any body is also related to relativistic momentum as:

E2 = (P c)2 + ( m0 c2)2 .......................(1)
Where, P = magnitude of momentum
For mass less particles m0 = 0
Hence E = p c
Relativistic Momentum
Figure: Depicting the concept of relativistic momentum

Impulse and Momentum

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The change of momentum is also termed as impulse. It is denoted by “I “.
Hence I = $\Delta$ P

$\frac{dp}{dt}$ = $\frac{d(mv)}{dt}$.
$\frac{dp}{dt}$ = m$\frac{dv}{dt}$ = ma

$\frac{dp}{dt}$
= F
Hence the above equations can be written as: dp = Fdt

So,

$\Delta$ P = F $\Delta$ T = P(final) – P(initial) ............................(1)and
I = $\Delta$ P = change in momentum ...........................(2)The S.I. unit of Impulse is Ns.
According to the law of conservation of momentum the momentum of the closed system of objects remains conserved given the fact no external force acts on it.
or
if no external force acts on a system in a particular direction then the total momentum of the system in the direction remains unchanged.
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Momentum Problems

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There can be many types of momentum problems. They can be based on the calculations of momentum, impulse, force etc.

The steps to be followed to solve the problems of Momentum are:
  1. If Momentum has to be calculated then calculate the initial and final velocities using the equations of motion.
  2. Calculate the Momentum by multiplying mass with velocity.
  3. To compute the impulses, calculate the force integrating it over time t1 and t2.

Mentioned below are few examples which help us understand Momentum:

Solved Examples

Question 1: A ball is thrown whose mass is 1 kg is traveling at 10 meters per second. Calculate its momentum?
Solution:
 
Given:
mass m = 1kg,
         Velocity v = 10 m/s.
Momentum p is given by: p = mv
                                        = (1 kg) $\times$ (10m/s)
                                        = 10 kg m/s.
 

Question 2: A body having 3 kg weight is traveling at 2m/s is subjected to Velocity of 10m/s.
Find:
        (a) Initial momentum
        (b) Final momentum
        (c) Change in momentum
        (d) Impulse
        (e) Calculate its magnitude if force acts for 1 s?
Solution:
 
Given mass m = 3kg,
        Initial Velocity Vi = 2m/s,
        Final Velocity Vf = 10m/s
(a) Initial momentum, pi = mvi = 3 $\times$ 2 = 6 kg m/s.
(b) Final momentum, pf = mvf = 3 $\times$ 10 = 30 kg m/s.
(c) Change in momentum, $\Delta$ p = pf - pi = 30 - 6 = 24kgm/s.
(d) Impulse, J = Ft = $\Delta$ p
                          = 24Ns.
(e)Magnitude of force is given by: F = $\frac{J}{t}$
                                                   = $\frac{24 Ns}{1 s}$
                                                   = 24N.
 

Impulse

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If the product of average force F, and the time it is exerted t, for which it acts is called Impulse. It is the net force acting on an object and is represented by the symbol “J”. The change in momentum of an object equals the impulse applied to it. 

For Example: A bat hitting a ball, the engine of a car accelerating it etc.

$\bar{dJ}$ = $\bar{F}$ dt

$\therefore$ $\bar{J}$ = $\int_{t_{1}}^{t_{2}}$ $\bar{F}$ dt

If the force does not vary with time, then
$\bar{J}$ = $\bar{F}$ $\Delta{t}$
The unit of Impulse are gcms-1 or kgms-1.
More topics in Momentum
Angular Momentum Conservation of Momentum
Elastic Collision Inelastic Collision
Related Topics
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