**The momentum equation is an equation that is used in open channel flow problems to determine unknown forces (F ) acting on the walls or bed in a control volume.** In comparison to the energy equation that deals with

*scalar*quantities such as mass (

*m*), pressure (

*P*), and velocity magnitude (

*V*), the

**momentum**equation deals with

*vector*quantities such as velocity vector and forces (

*F*).

## Momentum Equation

**Newton’s second law of conservation of momentum is as follows:**

**(1) ∑Fx=dM⃗ dt**

The conservation of momentum for a control volume asserts that the sum of all external forces acting on the volume plus the net rate of momentum entering the volume (momentum flux) equals the rate of momentum buildup.

In the x-direction, this is a vector equation. The fluid mass times the velocity vector in the flow direction equals the momentum flux (MV). There will be three forces considered: pressure, gravity, boundary drag, or friction force.

### Pressure forces:

The illustration below depicts an irregular cross-section in general. The total pressure force is the integral of the pressure-area product over the cross-section, and the pressure distribution is assumed to be hydrostatic (pressure varies linearly with depth). The pressure at any position, according to Shames (1962), can be written as:

**(2) Fp=∫h0ρg(h−y)T(y)dy**

The force at the control volume’s upstream end can be expressed as Fp if Fp is the pressure force in the x-direction at the control volume’s midpoint.

**(3) Fp−∂Fp∂xΔx2**

**(4) Fp+∂Fp∂xΔx2**

As a result, the total pressure forces for the control volume can be stated as:

**(5) FPn=∣∣∣FP−∂FP∂xΔx2∣∣∣−∣∣∣FP+∂FP∂xΔx2∣∣∣+FB**

The net pressure force for the control volume is FPn, and the force put on the fluid by the banks in the x-direction is FB. This can be summarised as follows:

**(6) FPn=−∂FP∂xΔx+FB**

Using Leibnitz’s Rule to differentiate (2) and then substituting in (6) yields:

**(7) FPn=−ρgΔx[∂h∂x∫h0T(y)dy+∫h0(h−y)∂T(y)∂xdy]+FB**

The cross-sectional area, A, is the first integral in (7). The second integral (multiplied by X) represents the fluid’s pressure force on the banks, which is identical in magnitude but opposite in direction to FB. As a result, the net pressure force can be expressed as:

**(8) FPn=−ρgA∂h∂xΔx**

### Gravitational force:

In the x-direction, the force of gravity on the fluid in the control volume is:

**(9) Fg=ρgAsinθΔx**

The angle formed by the channel invert with the horizontal is seen here. For natural rivers, sintan=Z0X, where z0 is the invert elevation, is modest. As a result, the gravitational force can be expressed as

**(10) Fg=−ρgA∂z0∂xΔx**

For negative bed slopes, this force will be positive.

Frictional forces between the channel and the fluid can be expressed in the following way:

**(11) Ff=−τ0PΔx**

P is the wetted perimeter, and 0 is the average boundary shear stress (force/unit area) acting on the fluid borders. With flow in the positive x-direction, the force operates in the negative x-direction, as shown by the negative sign. 0 can be stated in terms of a drag coefficient, CD, using dimensional analysis:

**(12) τ0=ρCDV2**

The drag coefficient may be connected to the Chezy coefficient, C, by the following:

**(13) CD=gC2**

The Chezy equation can also be written as:

**(14) V=CRSf−−−−√**

Substituting (12), (13), and (14) into (11) and simplifying results in the following boundary drag force expression:

**(15) Ff=−ρgASfΔx**

where Sf is the friction slope, which is positive for flow in the positive x-direction and negative for flow in the negative x-direction. The friction slope must be proportional to the flow and stage of the process.

The Manning and Chezy friction formulae have traditionally been utilized. The Manning equation is utilized in HEC-RAS because it is widely used in the United States. The Manning equation looks like this:

**(16) Sf=Q|Q|n22.208R4/3A2**

### Momentum flux:

Only the momentum flux remains after the three force terms have been defined. The flux into the control volume can be expressed as follows:

**(17) ρ[QV−∂QV∂xΔx2]**

as well as the flux leaving the volume, which can be written as:

**(18) ρ[QV+∂QV∂xΔx2]**

As a result, the control volume’s net rate of momentum (momentum flux) is:

**(19) −ρ∂(QV)∂xΔx**

Because the fluid’s momentum in the control volume equals Qx, the rate of momentum buildup can be stated as:

**(20) ∂∂t(ρQΔx)=ρΔx∂Q∂t**

**Reiterating the idea of momentum conservation:**

The rate of accumulation of momentum is equal to the net rate of momentum (momentum flux) entering the volume (19) plus the sum of all external forces acting on the volume [(8) + (10) + (15)]. (20). Hence:

**(21) ρΔx∂Q∂t=−ρ∂(QV)∂xΔx−ρgA∂h∂xΔx−ρgA∂z0∂xΔx−ρgASfΔx**

The water surface elevation, z, is equal to z0+h. Therefore:

**(22) ∂z∂x=∂h∂x+∂z0∂x**

where z/x is the slope of the water surface The final version of the momentum equation is obtained by substituting (22) into (21), dividing through by x, and moving all terms to the left:

**(23) ∂Q∂t+∂(QV)∂x+gA(∂z∂x+Sf)=0**

## Summary:

The conservation of momentum for a control volume is the sum of all external forces acting on the volume plus the net rate of momentum entering the volume (momentum flux). In the x-direction, this is a vector equation. There will be three forces considered: pressure, gravity, and boundary drag. The Manning and Chezy friction formulae have traditionally been utilized to work out how this happens.

## Momentum:

In Newtonian physics, linear momentum, often known as translational momentum or simply momentum, is the product of an object’s mass and velocity. It’s a magnitude and direction two-dimensional vector quantity.)If an object’s mass is m and its velocity is v (both vector quantities), the object’s momentum is p. The kilogram meter per second (kgm/s), which is equivalent to the newton-second in the International System of Units (SI), is the unit of measurement for momentum.

Newton’s second law of motion states that the rate of change of a body’s momentum is equal to the net force applied to it. Momentum is a conserved quantity in any inertial frame, which means that if a closed system is not influenced by external forces, its total linear momentum remains constant.

Momentum is maintained in special relativity, electrodynamics, quantum mechanics, quantum field theory, and general relativity (with a modified formula) (in a modified version). It’s a manifestation of translational symmetry, which is one of space and time’s fundamental symmetries.

Classical mechanics’ advanced formulations, such as Lagrangian and Hamiltonian mechanics, allow for the selection of coordinate systems that include symmetries and constraints. The preserved quantity in these systems is generalized momentum, which is not the same as the kinetic momentum mentioned above.

In quantum physics, the concept of generalized momentum is transformed into an operator on a wave function. The Heisenberg uncertainty principle connects the momentum and position operators. A momentum density can be defined in continuous systems such as electromagnetic fields, fluid dynamics, and deformable bodies, and equations such as the Navier–Stokes equations for fluids and the Cauchy momentum equation for deformable solids or fluids result from a continuum version of the conservation of momentum.

### Gaining Momentum

You’re probably familiar with the term “momentum.” You’ll frequently hear that something is gaining or acquiring traction. It could be a real moving object or something more metaphorical, such as a sports team.

But have you ever stopped to consider what the term “momentum” actually means?

Momentum is the quantity of motion of a moving body.In general, the more momentum a moving thing has, the more difficult it is to stop it.

This is why the term is often employed in a metaphorical sense, as in the case of a sports team. It indicates that the squad is on a roll (usually, a winning streak) and is growing as a result. The other teams will find it more difficult to stop the team from gathering momentum.

### Linear Momentum

We all know that momentum is the amount of motion a moving body has, but what does that mean? Consider a baseball being thrown in a straight path through the air to get a better understanding of this.

When you catch a baseball, you feel the ball’s momentum transfer to you. When you catch the ball, it will most likely push your hand back towards you. As it transfers its momentum to you, the more momentum the ball has, the more it will push your hand back.

Assume you’re getting hit by two baseballs. One is going 50 mph and the other is going 150 mph. Even if you manage to catch that 150 mph ball, it could knock you down. Stopping the 150 mph ball will take more work than stopping the 50 mph ball.

As a result, velocity is an extremely crucial part of the momentum. However, momentum is more than just that.Imagine two 50 mph balls being fired at you. The first is a baseball, whereas the second is a bowling ball. Trying to stop the bowling ball is generally not a good idea. Even after it touches you, it will continue to travel.

What makes the bowling ball so difficult to stop when both balls are going at the same speed? It’s because it’s more substantial. It has a greater bulk. As a result, mass is a crucial part of the momentum.

In physics, momentum is defined as the multiplication of mass and velocity, as seen in the following equation: p = m * v

• m Equals mass

• p = momentum

• v stands for velocity.

When it comes to momentum units, we don’t have a unique symbol for them. Instead, it is just the product of the kilogram (kg) standard unit of mass and the meter per second (m/s) standard unit of velocity. The standard unit of momentum is kilograms times meters per second (kg m/s).

### Dependence on the reference frame

Momentum is a quantifiable quantity whose measurement is determined by the frame of reference. For example, if an airplane of mass m kg is traveling at 50 m/s through the air, its momentum can be computed as 50m kg.m/s.

When flying into a 5 m/s headwind, the aircraft’s speed relative to the Earth’s surface is only 45 m/s, and its momentum is computed to be 45 m kg.m/s. Both of the calculations are valid. Any change in momentum will be determined to be consistent with the applicable physics laws in both frames of reference.

In a stationary frame of reference, suppose a particle is at point x. The position (represented by a primed coordinate) changes with time from the perspective of another frame of reference traveling at a uniform speed u.

A Galilean transformation is what this is known as. If the particle moves at dx/DT = v in the first frame of reference, it moves at dx/DT = v in the second.The accelerations are the same since you does not change:

In both reference frames, momentum is thus conserved. Furthermore, Newton’s second law remains unchanged in both frames as long as the force has the same form. This condition is satisfied by forces like Newtonian gravity, which is based solely on the scalar distance between objects.

Newtonian relativity, often known as Galilean invariance, refers to the independence of reference frames.A change in reference frame can often make motion calculations easier. In a collision between two particles, for example, a reference frame can be set where one particle starts at rest.

The center of mass frame - one that moves with the center of mass – is another often-used reference frame. In this frame, the overall momentum is zero.

## Summary:

The rate of change of a body’s momentum is equal to the net force applied to it, according to Newton’s second law of motion. In special relativity, electrodynamics, quantum mechanics, and general relativity, momentum is conserved.In quantum physics, momentum is transformed into an operator on a wave function. Momentum is the quantity of motion of a moving body.The more momentum a thing has, the more difficult it is to stop it. This is why the term is often employed in a metaphorical sense, such as when a sports team is on a winning streak.

### Application to collisions

The law of conservation of momentum can be used to determine the momentum of a coalesced body when two particles with known momentum collide and coalesce. If the two particles separate as a result of the collision, the law is insufficient to estimate each particle’s momentum.

The law can be used to determine the momentum of the other particle if the momentum of one particle after the collision is known.The collision is called an elastic collision if the energy is conserved; otherwise, it is called an inelastic collision.

#### Elastic collisions

An elastic collision is one in which no kinetic energy is converted to heat or another form of energy. Perfect elastic collisions can happen when particles do not touch each other, such as in atomic or nuclear scattering, where electric repulsion keeps the objects apart.

A fully elastic collision can also be interpreted as a slingshot maneuver of a satellite around a planet. Due to their great stiffness, a collision between two pool balls is a nice example of a nearly completely elastic collision, yet there is always some dissipation when bodies collide.

Velocities in one dimension, along a line running through the bodies, can be used to describe a head-on elastic collision between two bodies. If the velocities are u1 and u2 before the impact and v1 and v2 afterward, the equations for momentum and kinetic energy conservation are as follows:

A change in the reference frame can make collision analysis easier. Assume there are two bodies of identical mass m, one stationary and the other approaching at a speed of v. The center of mass is traveling at v/2, and both bodies are traveling in the same direction.

Because of the symmetry, both must be traveling away from the center of mass at the same speed after the impact. When we add the speed of the center of mass to both, we find that the moving body has come to a halt, while the other is traveling away at v.

The velocities of the bodies have swapped. A transition to the center of the mass frame brings us to the same conclusion regardless of the velocities of the bodies. As a result, gives the final velocities.

When the beginning velocities are known, the final velocities can be calculated using. If one body has a significantly higher mass than the other, the impact will not affect its velocity, whereas the other body will see a significant change.

#### Inelastic collisions

Some of the kinetic energy of colliding bodies is transformed into various kinds of energy in an inelastic collision (such as heat or sound). Traffic collisions, where the consequence of kinetic energy loss may be observed in vehicle damage; electrons losing some of their energy to atoms (as in the Franck–Hertz experiment); and particle accelerators, where kinetic energy is turned into mass in the form of new particles.

Both bodies have the same motion following a perfectly inelastic collision (such as an insect hitting a windshield). Velocities in one dimension, along a line running through the bodies, can be used to simulate a head-on inelastic collision between two bodies.

In a perfectly inelastic collision, if the velocities are u1 and u2 before the collision, both bodies will travel with velocity v after the impact.The equation that expresses momentum conservation is: If one body is immobile, to begin with (for example), the conservation of momentum equation is so

In a different circumstance, if the frame of reference moves at a final velocity, the objects will be brought to a complete stop by a perfectly inelastic collision, and all of the kinetic energy will be transformed into other kinds of energy. The initial velocities of the bodies would have to be non-zero in this case, or the bodies would have to be massless.

The coefficient of restitution CR, defined as the ratio of the relative velocity of separation to the relative velocity of approach, is one measure of collision inelasticity. This measure can be simply applied to a ball bouncing off a solid surface by utilizing the following formula:

The momentum and energy equations also apply to object motions that start together and eventually separate. A chain reaction, for example, turns chemical, mechanical, or nuclear potential energy into kinetic energy, acoustic energy, and electromagnetic radiation, resulting in an outbreak.

## Summary:

The law of conservation of momentum can be used to determine the momentum of coalesced bodies. An elastic collision is one in which no kinetic energy is converted to heat or another form of energy. A fully elastic collision can also be interpreted as a slingshot maneuver of a satellite around a planet. A change in the reference frame can make collision analysis easier. Assume there are two bodies of identical mass m, one stationary and the other approaching at a speed v. A transition to the center of mass frame brings us to the same conclusion regardless of the velocities of the bodies.

## Frequently Asked Questions:

**Following are the questions related to this keyword**

### 1: What is the equation for momentum?

**p = m v. The equation shows that momentum is proportional to the object’s mass (m) and velocity (v).** As a result, the larger the mass or velocity of an item, the greater its momentum. The momentum of a large, fast-moving item is larger than that of a smaller, slower object.

### 2: How do you write and solve equations in momentum?

Symbolically, linear momentum p is defined as **p = mv, where m is the mass of the system and v is its velocity.kg m/s is the SI unit for momentum. The net external force equals the change in momentum of a system divided by the duration during which it changes, according to Newton’s second law of motion in terms of momentum.**

### 3: How do you find final momentum?

**The final momentum would be (4+6) equal to the mass of both balls multiplied by the final velocity (vf).** The conservation of momentum allows us to find vf; the sum of the initial momentum values must equal the ending momentum.

### 4: What is momentum in physics?

**Momentum is defined as the product of a particle’s mass and velocity.In the sense that it has both a magnitude and a direction, momentum is a vector quantity.** According to Isaac Newton’s second equation of motion, the particle’s temporal rate of change of momentum is equal to the force applied to it. For more information, see Newton’s laws of motion.

### 5: How do you find momentum with force and time?

**You may calculate the change in momentum of an object by knowing the amount of force applied and the length of time it was applied.** The fact that force is the rate at which momentum varies to time (F = dp/DT) connects them. If p = mv and m is constant, F = dp/DT = m*dv/DT = ma is the result.

### 6: What are mass and momentum?

**The momentum of an object is equal to its mass multiplied by its velocity in terms of an equation.Momentum is equivalent to mass times velocity.** In physics, the symbol for the quantity momentum is lower case p.Thus, p = m • v can be written as p = m • v, where m denotes mass and v denotes velocity.

### 7: How can a tennis ball have the same momentum as a bowling ball?

If there are any changes after that, they must be the same, or the overall momentum will not be constant. As a result, **the tennis ball will change velocity more but has less mass, allowing it to maintain the same momentum change as the bowling ball, which will not change velocity much but has a larger mass.**

### 8: How do you find velocity after a collision?

**The velocity change in a collision is always calculated by subtracting the beginning velocity from the final velocity.** If an object is going in one direction before colliding with another item and then bounces or changes direction, its velocity after the collision is in the opposite direction.

### 9: Which has more momentum a bullet or a car?

**Because of its greater mass, the automobile has more momentum.** The amount of momentum an object has is also affected by its velocity. An arrow launched from a bow, for example, has considerable momentum because, despite its little mass, it travels at a rapid velocity.

### 10: What are the two quantities needed to calculate an object’s momentum?

**The amount of material moving and the speed at which it is travelling are the two parameters that define how much momentum an item has.** Momentum is influenced by mass and velocity. The momentum of an object is equal to its mass multiplied by its velocity in terms of an equation.

## Conclusion:

The momentum of a large, fast-moving item is larger than that of a smaller, slower object. Momentum is defined as the product of a particle’s mass and velocity.Momentum is equal to the force applied on the particle, according to Newton’s second law of motion. The amount of momentum that an object possesses is determined by two factors. These are the amount of material moving and the speed at which it is traveling.

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