3-Dimensional Position and Motion

Alexander Haniotis Fall 2025

Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

The Main Idea

Objects in our world move through three-dimensional space, which we describe using the [math]\displaystyle{ \hat{x}, \hat{y}, \hat{z} }[/math] coordinate directions. A key idea in physics is that motion along each axis is independent of motion along the others. This allows us to treat three-dimensional motion as three separate one-dimensional problems.

However, solving problems using vector notation is usually faster and more elegant because vectors combine these three components into a single mathematical object. For example, a velocity vector can be written as:

[math]\displaystyle{ \vec{v} = \frac{\Delta \vec{r}}{\Delta t} = \left\langle \frac{\Delta x}{\Delta t}, \frac{\Delta y}{\Delta t}, \frac{\Delta z}{\Delta t} \right\rangle }[/math]

The foundation of all three-dimensional motion is the position vector, [math]\displaystyle{ \vec{r} = \langle x, y, z \rangle }[/math], because quantities such as velocity and acceleration are defined using changes in position:

Velocity: [math]\displaystyle{ \vec{v} = \frac{\Delta \vec{r}}{\Delta t} }[/math]

Acceleration: [math]\displaystyle{ \vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\Delta \vec{r}}{\Delta t^2} }[/math]

By building motion from the position vector and its derivatives, we can describe how objects move in three-dimensional space with clarity and precision.

A Mathematical Model

Kinematic Equations

The kinematic equations describe motion of an object when the net force, and therefore the acceleration, is constant. In three dimension, each equations applies to all three components of the motion.

[math]\displaystyle{ d= d_0+ v_0t+ }[/math] [math]\displaystyle{ at^2 \over\ 2 }[/math]

[math]\displaystyle{ v= v_0+at }[/math]

[math]\displaystyle{ v^2= v_0^2 + 2a(d-d_0) }[/math]

Momentum Principle

When force changes with time or position, the kinematic equations are no longer valid. In those cases, the momentum principle provides the correct model.

A Computational Model

Three–dimensional motion is especially powerful in computing, given the increased complexity opposed to working with one or two dimensions. Using a computational model, we can update an object’s position step–by–step in time using vector quantities like position, velocity, momentum, and net force. GlowScript VPython lets us visualize these updates in real time in a web browser.

In each of the following examples, the motion is computed iteratively with a small time–step dt. This mirrors the momentum principle:

[math]\displaystyle{ \vec{p}{f} = \vec{p}{i} + \vec{F}_{\text{net}} \Delta t }[/math]

and the update of position from the velocity.

Constant–Velocity Motion in 3D

This simple model shows a particle moving with constant velocity in three dimensions.

Web VPython 3.2

scene.title = "Constant–velocity motion in 3D"
scene.caption = "The sphere moves with constant velocity and leaves a trail."

ball = sphere(pos=vector(-5, 0, 0), radius=0.2, color=color.cyan, make_trail=True)
v = vector(1.0, 0.6, 0.3)   # constant velocity (m/s)

dt = 0.01
while True:
    rate(100)
    ball.pos = ball.pos + v*dt

Projectile Motion with Gravity

Here we add a constant downward acceleration to model gravity. The motion is still three–dimensional, but the force is only in the vertical direction.

Web VPython 3.2

scene.title = "3D projectile under gravity"
scene.caption = "The red sphere is launched with an initial velocity and moves under gravity."

ball = sphere(pos=vector(0, 0, 0), radius=0.2, color=color.red, make_trail=True)

g = vector(0, -9.8, 0)        # gravitational field (m/s^2)
v = vector(6.0, 8.0, 2.0)     # initial velocity (m/s)
dt = 0.01

while ball.pos.y >= 0:
    rate(200)
    v = v + g*dt
    ball.pos = ball.pos + v*dt

Examples

Here are a few examples:

Simple

At t = 10.0 seconds a mass of 3 kg has velocity of [math]\displaystyle{ \langle 12, 27, −8 \rangle }[/math] m/s. At t = 10.1 seconds its velocity was [math]\displaystyle{ \langle 24, 19, 22 \rangle }[/math] m/s. What was the average (vector) net force acting on the object?

[math]\displaystyle{ \vec{v}= \vec{v}_0+\vec{a}t }[/math]

[math]\displaystyle{ \vec{v}-\vec{v}_0 \over\ t }[/math] [math]\displaystyle{ = \vec{a} }[/math]

Remember Newton's second law [math]\displaystyle{ \vec{F}_{net}= m\vec{a} }[/math]

[math]\displaystyle{ m(\vec{v}-\vec{v}_0) \over\ t }[/math] [math]\displaystyle{ = m(\vec{a}) }[/math]

[math]\displaystyle{ m(\vec{v}-\vec{v}_0) \over\ t }[/math] [math]\displaystyle{ = \vec{F}_{net} }[/math]

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

[math]\displaystyle{ \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} }[/math]

Middling

A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a string. You hold the toy such that the feathers hang suspended from the string when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is [math]\displaystyle{ \langle −0.02, −0.01, −0.02 \rangle }[/math] kg · m/s, and the moving ball is at location [math]\displaystyle{ \langle −0.2, −0.61, 0 \rangle }[/math] m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step [math]\displaystyle{ \Delta t }[/math] of 0.1 s?

[math]\displaystyle{ d= d_0+ v_0t }[/math]

remember the definition of velocity in relation to momentum: [math]\displaystyle{ \vec{p} \over\ m }[/math] [math]\displaystyle{ = \vec{v} }[/math]

[math]\displaystyle{ d= d_0+ }[/math] [math]\displaystyle{ \vec{p}_{0} \over\ m }[/math] [math]\displaystyle{ t }[/math]

Difficult

An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was [math]\displaystyle{ \langle 4.22, 2.45, −9.63 \rangle }[/math] m.

At t = 1.59 s, the position was [math]\displaystyle{ \langle 4.26, 2.37, −10.35 \rangle }[/math] m.

Second Interval:

At t = 3.56 s, the position was [math]\displaystyle{ \langle 8.09, 6.18, -58.35 \rangle }[/math] m.

At t = 3.59 s, the position was [math]\displaystyle{ \langle 8.17, 6.37, -59.07 \rangle }[/math] m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

[math]\displaystyle{ {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} }[/math]

[math]\displaystyle{ {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} }[/math]

[math]\displaystyle{ {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle }[/math]

(b) Begin with the definition of momentum:

[math]\displaystyle{ {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} }[/math]

[math]\displaystyle{ {\vec{p}_{avg,2}}= {m{\vec{v}_{f}-\vec{v}_{i} \over\ \Delta t}} }[/math]

[math]\displaystyle{ {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle }[/math]

(c) Begin from a fundamental principle

[math]\displaystyle{ {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} }[/math]

[math]\displaystyle{ {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} }[/math]

[math]\displaystyle{ \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} }[/math]

Connectedness

1. How is this topic connected to something that you are interested in?

Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.

1. How is it connected to your major?

I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.

1. Is there an interesting industrial application?

Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which demand an ability to visualize parts in three dimensions.

History

Compared to Newtonian physics, the field of kinematics is a relatively recent exploration. André-Marie Ampère wrote in his 1834 essay Essai sur la philosophie des sciences about the need for a field of science which analyzes motion independent of the forces. Work continued throughout the nineteenth century under Franz Reuleaux who is considered the father of modern kinematics. In 1666, Newton formulated early versions of his three laws of motion, of which the firstl aw describes the momentum principle. Two decades later, he would publish Principia which is often cited as one of greatest scientific books ever written.

See also

Further Reading

http://kmoddl.library.cornell.edu/what.php

References

http://farside.ph.utexas.edu/teaching/301/lectures/node33.html http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Newton.html http://kmoddl.library.cornell.edu/what.php