2.4 Motion in Space: Velocity and Acceleration

Vector Functions and Motion

Vector functions let you describe how an object moves through 3D space by tracking its position, velocity, and acceleration as functions of time. This is where the calculus of derivatives meets physics: you differentiate to go from position to velocity to acceleration, and integrate to go the other direction.

Position, Velocity, and Acceleration Vectors

These three vectors form a chain of derivatives. Each one is the derivative of the one before it.

Position $\mathbf{r}(t)$ tells you where a particle is at time $t$:

$\mathbf{r}(t) = x(t)\,\mathbf{i} + y(t)\,\mathbf{j} + z(t)\,\mathbf{k}$

Velocity $\mathbf{v}(t)$ is the rate of change of position. You get it by differentiating each component:

$\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = x'(t)\,\mathbf{i} + y'(t)\,\mathbf{j} + z'(t)\,\mathbf{k}$

The velocity vector is always tangent to the curve at the particle's current location. This is a key geometric fact.

Acceleration $\mathbf{a}(t)$ is the rate of change of velocity:

$\mathbf{a}(t) = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2} = x''(t)\,\mathbf{i} + y''(t)\,\mathbf{j} + z''(t)\,\mathbf{k}$

So the process is straightforward: differentiate component by component, just like you would with a single-variable function, but do it for each coordinate separately.

Speed vs. Velocity

Velocity is a vector: it has magnitude and direction. Speed is a scalar: it's just how fast you're going, with no directional information.

$\text{speed} = |\mathbf{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}$

This distinction matters. An object moving in a circle at constant speed still has nonzero acceleration because its direction is changing. Speed only tells you the magnitude part; velocity captures the full picture.

Similarly, the magnitude of acceleration is:

$|\mathbf{a}(t)| = \sqrt{(x''(t))^2 + (y''(t))^2 + (z''(t))^2}$

Types of Motion Analysis

Projectile Motion

Projectile motion is what happens when the only force acting on an object is gravity (no air resistance). The horizontal and vertical components separate cleanly:

• Horizontal: $x(t) = x_0 + (v_0 \cos\theta)\, t$ (constant velocity, no acceleration)
• Vertical: $y(t) = y_0 + (v_0 \sin\theta)\, t - \frac{1}{2}g\,t^2$ (constant downward acceleration $g \approx 9.8 \text{ m/s}^2$)

The position vector is $\mathbf{r}(t) = x(t)\,\mathbf{i} + y(t)\,\mathbf{j}$, and the acceleration vector is simply $\mathbf{a}(t) = -g\,\mathbf{j}$. Notice that acceleration is entirely vertical.

Circular Motion

For uniform circular motion (constant speed around a circle of radius $r$), a common parametrization is:

$\mathbf{r}(t) = r\cos(\omega t)\,\mathbf{i} + r\sin(\omega t)\,\mathbf{j}$

where $\omega$ is the angular velocity, related to linear speed by $\omega = \frac{v}{r}$.

If you differentiate twice, you'll find the acceleration points inward toward the center of the circle with magnitude:

$a_c = \frac{v^2}{r} = \omega^2 r$

This is centripetal acceleration. Even though speed is constant, the direction keeps changing, so acceleration is nonzero.

Tangential and Normal Components of Acceleration

Any acceleration vector can be decomposed into two perpendicular parts:

• Tangential component $a_T = \frac{d|\mathbf{v}|}{dt}$ controls changes in speed (speeding up or slowing down along the path)
• Normal component $a_N = \frac{|\mathbf{v}|^2}{\rho}$ controls changes in direction (how sharply the path curves), where $\rho$ is the radius of curvature

The total acceleration is:

$\mathbf{a} = a_T\,\mathbf{T} + a_N\,\mathbf{N}$

where $\mathbf{T}$ is the unit tangent vector and $\mathbf{N}$ is the unit normal vector. This decomposition is powerful because it separates "how much is the object speeding up?" from "how much is the object turning?"

A useful computational formula: $a_T = \frac{\mathbf{v} \cdot \mathbf{a}}{|\mathbf{v}|}$ and $a_N = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|}$. These let you find the components directly from $\mathbf{v}$ and $\mathbf{a}$ without needing $\mathbf{T}$ and $\mathbf{N}$ explicitly.

Applications of Vector Functions in Motion

• Planetary orbits: Kepler's laws describe elliptical paths; vector functions model these trajectories and the varying speed of a planet as it moves closer to or farther from the Sun.
• Engineering design: Roller coasters and road curves are designed using curvature and normal acceleration to keep forces within safe limits for passengers.
• Satellite mechanics: Orbital velocity and escape velocity calculations rely on balancing gravitational acceleration with the satellite's velocity vector.
• Sports physics: Ball trajectories (a kicked soccer ball, a golf drive) combine projectile motion with spin and drag effects that modify the basic parabolic path.