Predicting Change in multiple dimensions

This page discusses the use of momentum to predict change in multi-dimensions and examples of how it is used.

Claimed by rbose7

The Main Idea

Just as in one dimension, the linear momentum (also known as translational momentum) of an object is the vector quantity equal to the product of the mass and velocity of an object. Unlike in one dimension, we must now consider the vectors components in each direction (see 3-Dimensional Position and Motion for more detail). Just as in 1 dimension, momentum is conserved in closed systems - that is, a system with no external forces acting upon it - which means that two objects colliding will have the same net momentum before and after the collision. We can apply these properties to all three dimensions and use momentum to predict the path an object will follow over time by observing the change in momentum just as we did in one-dimension.

A Mathematical Model

This change in momentum is shown by the formula:

[math]\displaystyle{ \Delta \vec{p} = \vec{p}_{f}-\vec{p}_{i} = m\vec{v}_{f}-m\vec{v}_{i} }[/math]

Or by relating it to force:

[math]\displaystyle{ \Delta \vec{p} = \vec{F} \Delta t }[/math]

Relation to Velocity

Given an object with velocity [math]\displaystyle{ \vec{v} = (v_x,v_y,v_z) }[/math] and mass [math]\displaystyle{ m }[/math], the object's momentum will be

[math]\displaystyle{ \vec{p} = m\vec{v} = m(v_x,v_y,v_z) = (m v_x,m v_y,m v_z) }[/math]

Relate by Force

Given the force:

[math]\displaystyle{ \vec{F} = (F_x,F_y,F_z) }[/math]

And change in time:

[math]\displaystyle{ \Delta t }[/math]

[math]\displaystyle{ \Delta p = \vec{F} \Delta t = (F_x,F_y,F_z) \Delta t= ( F_x \Delta t,F_y \Delta t ,F_z \Delta t) }[/math]

[math]\displaystyle{ \vec{p}_f = \vec{p}_{i} + \Delta p = \vec{p}_i + (F_x \Delta t,F_y \Delta t,F_z \Delta t) }[/math]

This can also be expressed as:

[math]\displaystyle{ \vec{F}_{net} = \frac {\text{d}\vec{p}} {\text{d}t} }[/math]

or:

[math]\displaystyle{ \Delta p = \int_{t_1}^{t_2} F(t)\, dt\,. }[/math]

Multiple Particles

If our system consists of multiple particles being acted upon by a net external force, we can use the same process to predict its evolution. The only difference is that we pretend the particles are just on large particle with its center at the center of mass.

Center of Mass:

In summation notation:

[math]\displaystyle{ \vec{r}_{cm} = \frac{\sum_i^n m_i\vec{r}_i}{\sum_i^n m_i} }[/math]

If you are not familiar with summation notation, a brief recap may be found here, but it essentially consists of adding the product of each mass with its respective position, then dividing by the total mass. Accordingly, we have in expanded notation

[math]\displaystyle{ (x_{cm},y_{cm},z_{cm}) = \frac{1}{m_{total}} (\sum_i^n m_ix_i,\sum_i^n m_iy_i,\sum_i^n m_iz_i) }[/math]

Performing unit analysis, we see that the masses cancel out and this is indeed a position. With these quantities we may perform the same computations as above, but instead of using [math]\displaystyle{ m, \vec{v}, }[/math] and [math]\displaystyle{ r }[/math], we use [math]\displaystyle{ m_{total} = \sum_i^n m_i }[/math],[math]\displaystyle{ \vec{v}_{cm} }[/math], and [math]\displaystyle{ \vec{r}_{cm} }[/math].

A Computational Model

The principles of iterative prediction in multiple dimensions are exactly the same as the Fundamentals of Iterative Prediction with Varying Force, except that as per the above the process must be performed with 3 times as much information. The most important difference is that the force in one direction may depend on the position and velocity in another - Oscillators in Multiple Dimensions will consider such cases - which requires the system to update each coordinate simultaneously, rather than treating it like three separate one dimensional problems.

A Generic 3D Simulator

Using Numpy

Below are models that use change in momentum to predict how particles move in specific situations (Click run to start simulation):

(If it does not work take the '?outputOnly=true' out of the url and try again)

A object with no net force on it

Below is a particle that has no net force and therefore moves at a constant velocity:

A object with no net force on it

A object with the force of gravity

Below is an object moving with gravity acting on it. Because gravity acts in the 'y' direction, the object's y component for velocity decreases:

A object with the force of gravity

Many Particles

Below are several objects moving with gravity acting on it, using calculations from center of mass (it is usually more accurate to apply calculations on each particle individually, but this is good for a big picture).

Many Particles

A object launched from a cliff

Below is an object launched with an initial velocity that has gravity acting on it. It loses velocity in the y direction due to gravity until it hits the ground:

A object launched from a cliff

Others

Projectile Motion

Examples

Simple

A ball of mass [math]\displaystyle{ 1 \; kg }[/math] has initial velocity [math]\displaystyle{ (3,4,0) \; m/s }[/math] in a pool of water, where the z-axis is the axis of gravity. The ball has exactly the density of water, so it buoyancy exactly cancels out gravity, and it only feels the force of drag due to the water. It feels linear drag, described by the equation [math]\displaystyle{ \vec{F}_d = -b\vec{v} }[/math], with [math]\displaystyle{ b = 0.1 \; kg/s }[/math]. Find the vector form of the drag force on the ball, and state its magnitude.

Solution

All we have to do is plug our values into the given equation, but we do need to get the vector right:

[math]\displaystyle{ \vec{F}_d = -b \vec{v} \rightarrow (F_x,F_y,F_z) = -b(v_x,v_y,v_z) }[/math]

The scalar term (the coefficient) will multiply out through the vector, so it results in

[math]\displaystyle{ (F_x,F_y,F_z) = -(0.1 \; kg/s)\cdot(3,4,0) \; m/s = (-0.3,-0.4,0) \; N }[/math]

Now, to find the magnitude we take a pythagorean sum:

[math]\displaystyle{ |\vec{F}_d| = \sqrt{F_x^2 +F_y^2+F_z^2} = \sqrt{0.25 \; N^2} = 0.5 \; N }[/math]

Note that the magnitude is strictly positive, and that the negatives are therefore coming from the direction of the vector. Although it's not asked for, note as well that it may be useful to determine the angle of the force with the x-axis, which we do by computing

[math]\displaystyle{ \theta = \arctan\biggr{(} \frac{F_y}{F_x}\biggr{)} = 53.13^\circ = 0.927 \; \text{rad} }[/math]

Middling

In the same situation as described above, compute the position (assuming the ball starts at the origin) after two time steps of [math]\displaystyle{ 0.1 \; s }[/math].

Solution

We have the first iterations force already worked out, so we compute the change in momentum:

<math> \Delta \vec{p} =

Difficult

Connectedness

Momentum is what some physicists call the fundamental principle. Almost everything in physics is based off this principle. This is why it is one of the most interesting parts of physics. From getting a few measurements, we have shown how we can predict the final destination of any particle. We can create models for real life examples, like finding how hard we have to kick a ball in a sport, where a ball will stop rolling on the ground and for many instances of projectile motion. It basically allows us to create mathematical models and predict the final destination of an object for almost any situation.

For my major, computer science, there are a few applications. For example, every computational model in the section above is coded using Python. A big part of computer science is modeling real life examples on a computer to mimic situations that would be otherwise impractical or impossible to set up. For example, the model we created of the electron and proton moving would take several expensive instruments to set up and visualize the two particles. For several other instances, using the principle of momentum and analyzing the change in momentum we can model several situations using coding that would otherwise not be possible.

History

Before Newton, French scientist and philosopher Descartes introduced the concept of momentum. He used the concept of momentum to describe how people moved when objects were thrown at them. He focused generally on the conservation of momentum when dealing with collisions. Newton's laws further expanded on the idea of conservation of momentum. The ideas that F = ma and the idea that for every action there is an equal and opposite reaction are the basis for many problems and concepts explained in this section. More information here:

http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html

https://en.wikipedia.org/wiki/Momentum#History_of_the_concept

See also

Linear Momentum

Projectile Motion

Kinematics

Momentum Principle

Fundamentals of Iterative Prediction with Varying Force

External Links

https://en.wikipedia.org/wiki/Momentum

https://en.wikibooks.org/wiki/General_Mechanics/Momentum

http://hyperphysics.phy-astr.gsu.edu/hbase/mom.html

http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html

References

https://en.wikipedia.org/wiki/Momentum

https://en.wikibooks.org/wiki/General_Mechanics/Momentum

http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html