Fluid Mechanics: The Momentum Equation
Fluid Mechanics: The Momentum Equation

Momentum

Momentum, sometimes also known as linear momentum, is defined as the product of an object’s mass and velocity. Alternatively, momentum is defined as ‘mass in motion’. Because all objects have mass, they have momentum if they move, which indicates their mass is moving.

Any change in the system’s mass or velocity results in a change in linear momentum. Linear momentum is proportional to an object’s mass and velocity. The following is a formula for linear momentum:

Momentum = mass × velocity

p=m×v

Here,

p= Linear Momentum

m= Mass

v= Velocity

From this, it can be deduced that the greater the object’s mass or velocity, the greater the linear momentum.

Because it has both a direction and a magnitude, momentum is a vector quantity. It moves in the same direction as velocity.

The SI unit of momentum is kg.m/s

Conservation of Momentum

The quantity called momentum that characterises motion in an isolated collection of objects never changes, according to the conservation of momentum principle; that is, the overall momentum of a system remains constant. When the mass of an object is multiplied by its velocity, momentum equals the force required to bring it to a complete stop in a given amount of time. A set of objects’ overall momentum is equal to the sum of their individual momenta. Because momentum is a vector that comprises both the direction and the amplitude of motion, the momenta of objects travelling in opposite directions might cancel out, resulting in a total sum of zero.

A rocket’s and its fuel’s total momentum are zero before launch. During launch, the downward momentum of the expanding exhaust gases just equals the upward momentum of the rising rocket, resulting in a constant total momentum of the system—in this case, zero. The sum of the two momentum before collision equals the sum of the two momentum after collision in a collision of two particles. When one particle loses in momentum, the other gains.

Formula of Conservation of Mass

Mathematically, Conservation of Mass can be given by:

Let us consider that two objects, a bowling ball and basketball are colliding with each other.

m1u1+m2u2=m1v1+m2v2

m1= Mass of bowling ball

m2= Mass of the basketball

u1= Initial velocity of the bowling ball

u2= Initial velocity of the basketball

v1= Final velocity of the bowling ball

v2= Final velocity of the basketball

Derivation of Conservation of Momentum

Newton’s third law states that when an object A produces a force on an object B, object B responds with a force of the same magnitude but opposite direction. The law of conservation of momentum was derived from this concept by Newton.

Let us Consider two particles A and B which collide with each other with masses are m1 and m2 with initial velocities u1and u2 respectively, and final velocities v1 and v2. The time of contact between A and B is given as t.

Conservation of Momentum Example

Consider the case of a balloon, where gas particles collide frequently with each other and the balloon’s walls; despite the fact that the particles themselves move faster and slower as they lose and gain momentum when they collide, the system’s total momentum remains constant.

As a result, the balloon does not change size when we add external energy by heating it. This raises the velocity of the particles, which increases their momentum, which increases the force imposed by them on the balloon’s walls.

Interesting Facts about Conservation of Momentum

There are four facts regarding momentum conservation that are really interesting:

  1. Because momentum is a vector variable, we must use vector addition to add the momenta of numerous bodies that make up a system. Consider a system consisting of two comparable items travelling in opposite directions at the same speed. The fact that the oppositely-directed vectors cancel out means that the system’s overall momentum is zero, despite the fact that both objects are moving.

  2. The use of conservation of momentum to study collisions is particularly intriguing. Because collisions usually happen quickly, the amount of time colliding objects spend interacting is limited. The impulse, F.∆t due to external forces such as friction during the collision is relatively minimal when the interaction period is short.

  3. Even with complex systems involving many components, measuring and keeping track of momentum is often simple. Consider the impact of two ice hockey pucks colliding. One of the pucks gets shattered into two pieces as a result of the contact. The impact will most likely not conserve kinetic energy, but momentum will.

We can still use conservation of momentum to comprehend the situation if we know the masses and velocities of all the pieces just after the impact. This is intriguing since, in this case, using conservation of energy would be almost difficult. It would be difficult to determine how much effort was put into shattering the puck.

  1. It’s exciting to observe collisions with “immovable” objects. Of course, nothing is actually immovable, even though some appear to be due to their enormous size. Consider a bouncy ball with mass m and a velocity v approaching a brick wall. It collides with the wall and bounces back at -v . The ball’s momentum has changed by 2mv ,since velocity moved from positive to negative, despite the fact that the wall is well anchored to the earth and does not move.

If momentum is conserved, then the momentum of the earth and wall also must have changed by 2mv . We just don’t notice this because the earth is so much heavier than the bouncy ball.

Conclusion

The phrase conservation refers to something that does not change in physics. This signifies that a conserved quantity’s variable in an equation remains constant over time. Its value is the same before and after an event.

In physics, there are numerous conserved quantities. They are frequently remarkably beneficial for making predictions in situations that would otherwise be extremely complex. There are three fundamental quantities in mechanics that are conserved. Momentum, energy, and angular momentum are the three. The term “conservation of momentum” is most commonly used to describe object collisions.

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