3.4 Motion in Space

Motion in three-dimensional space takes us beyond the flat plane. We're now tracking objects as they zip through the air, curve around corners, and follow complex paths. This opens up a whole new world of possibilities for understanding movement.

Velocity and acceleration vectors help us describe these 3D motions precisely. We'll see how breaking these vectors into components lets us analyze both speed changes and direction changes separately, giving us a clearer picture of what's really happening.

Motion in Three-Dimensional Space

Velocity and acceleration in 3D space

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• Velocity vector $\vec{v}(t)$ represents the rate of change of position with respect to time in three-dimensional space
  • Defined as the first derivative of the position vector $\vec{r}(t)$ with respect to time: $\vec{v}(t) = \frac{d\vec{r}(t)}{dt} = \langle \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \rangle$
  • Velocity vector components $\frac{dx}{dt}$, $\frac{dy}{dt}$, and $\frac{dz}{dt}$ represent the rates of change of position along the $x$, $y$, and $z$ axes, respectively
• Magnitude of the velocity vector $|\vec{v}(t)|$ represents the speed of the particle at any given time
  • Calculated using the Pythagorean theorem in three dimensions: $|\vec{v}(t)| = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2}$
  • Scalar quantity that measures the rate at which the particle covers distance along its path (speed of a car, velocity of a projectile)
• Acceleration vector $\vec{a}(t)$ represents the rate of change of velocity with respect to time in three-dimensional space
  • Defined as the second derivative of the position vector $\vec{r}(t)$ or the first derivative of the velocity vector $\vec{v}(t)$ with respect to time: $\vec{a}(t) = \frac{d\vec{v}(t)}{dt} = \langle \frac{d^2x}{dt^2}, \frac{d^2y}{dt^2}, \frac{d^2z}{dt^2} \rangle$
  • Acceleration vector components $\frac{d^2x}{dt^2}$, $\frac{d^2y}{dt^2}$, and $\frac{d^2z}{dt^2}$ represent the rates of change of velocity along the $x$, $y$, and $z$ axes, respectively
• Magnitude of the acceleration vector $|\vec{a}(t)|$ represents the rate at which the particle's velocity is changing at any given time
  • Calculated using the Pythagorean theorem in three dimensions: $|\vec{a}(t)| = \sqrt{\left(\frac{d^2x}{dt^2}\right)^2 + \left(\frac{d^2y}{dt^2}\right)^2 + \left(\frac{d^2z}{dt^2}\right)^2}$
  • Scalar quantity that measures how quickly the particle's velocity is changing (acceleration of a falling object, deceleration of a braking vehicle)
• Velocity and acceleration vectors are always tangent to the particle's path at any given point, indicating the instantaneous direction of motion and change in velocity, respectively (velocity and acceleration of a roller coaster at a specific point on the track)

Tangential vs normal acceleration components

• Tangential component of acceleration $a_T$ represents the rate of change of speed along the particle's path
  • Calculated as the derivative of the magnitude of the velocity vector with respect to time: $a_T = \frac{d|\vec{v}|}{dt}$
  • Tangential unit vector $\hat{u}_T$ points in the direction of motion, parallel to the velocity vector
  • Positive $a_T$ indicates increasing speed, while negative $a_T$ indicates decreasing speed (accelerating or decelerating car)
• Normal component of acceleration $a_N$ represents the rate of change of direction of the particle's path
  • Calculated using the formula $a_N = \frac{v^2}{\rho}$, where $v$ is the speed and $\rho$ is the radius of curvature of the path
  • Normal unit vector $\hat{u}_N$ points toward the center of curvature, perpendicular to the velocity vector
  • Always points inward for curved paths, causing a change in the direction of motion (car turning a corner, planets orbiting the sun)
  • This inward-pointing acceleration is also known as centripetal acceleration
• Total acceleration vector $\vec{a}$ is the vector sum of the tangential and normal components: $\vec{a} = a_T \hat{u}_T + a_N \hat{u}_N$
  • Magnitude of the total acceleration: $|\vec{a}| = \sqrt{a_T^2 + a_N^2}$
  • Direction of the total acceleration depends on the relative magnitudes of $a_T$ and $a_N$ (roller coaster at the bottom of a loop, where both tangential and normal components are present)

Planetary Motion and Kepler's Laws

Kepler's laws for celestial mechanics

1. Kepler's first law (law of ellipses) states that planets move in elliptical orbits with the sun at one focus
   • Equation of an ellipse in standard form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the semi-major and semi-minor axes, respectively
   • Eccentricity $e$ of an ellipse measures its deviation from a circle: $e = \sqrt{1 - \frac{b^2}{a^2}}$ (Earth's orbit with $e \approx 0.0167$, Pluto's orbit with $e \approx 0.2488$)
2. Kepler's second law (law of equal areas) states that a line segment joining a planet and the sun sweeps out equal areas during equal intervals of time
   • Consequence of the conservation of angular momentum in the absence of external torques
   • Planets move faster when closer to the sun and slower when farther away (Earth's orbital speed varies from $\approx 30.3$ km/s at perihelion to $\approx 29.3$ km/s at aphelion)
3. Kepler's third law (law of periods) states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit
   • Mathematically expressed as $\frac{T^2}{a^3} = \frac{4\pi^2}{GM}$, where $T$ is the orbital period, $a$ is the semi-major axis, $G$ is the gravitational constant, and $M$ is the mass of the central body (sun)
   • Allows for the determination of relative distances of planets from the sun based on their orbital periods (Earth's orbital period of 1 year and semi-major axis of 1 AU, Jupiter's orbital period of $\approx 11.86$ years and semi-major axis of $\approx 5.20$ AU)

Gravitational Forces and Orbital Dynamics

• Gravitational force between two bodies is described by Newton's law of universal gravitation: $F = G\frac{m_1m_2}{r^2}$
• This force is responsible for keeping planets in orbit around the sun and moons around planets
• Escape velocity is the minimum speed needed for an object to break free from a planet's gravitational field without further propulsion
• Angular momentum is conserved in orbital motion, leading to Kepler's second law and affecting orbital shapes