Motion in Space: Velocity, Acceleration, and Projectile Motion

This study guide covers the application of vector-valued functions to describe the kinematics of particles moving through two-dimensional and three-dimensional space.

Learning Objectives

By the end of this module, you should be able to:

• Calculate the velocity, acceleration, and speed vectors for a particle given its position function $\mathbf{r}(t)$.
• Interpret the physical meaning of the tangent vector as it relates to the direction of motion.
• Solve problems involving projectile motion by applying initial conditions and gravitational constants.
• Decompose acceleration into its tangential and normal components ($a_T$ and $a_N$).

Key Terms & Glossary

• Position Vector ($\mathbf{r}(t)$): A vector-valued function representing the location of an object at time $t$.
• Velocity Vector ($\mathbf{v}(t)$): The first derivative of position with respect to time; it is always tangent to the path of motion.
• Speed ($v(t)$): The magnitude of the velocity vector, a scalar quantity: $\|\mathbf{v}(t)\|$.
• Acceleration Vector ($\mathbf{a}(t)$): The derivative of the velocity vector; represents the rate of change of velocity.
• Projectile Motion: The motion of an object thrown or projected into the air, subject only to the acceleration of gravity.

The "Big Idea"

In single-variable calculus, we study motion along a line. In Motion in Space, we extend these concepts using vector calculus. The "Big Idea" is that the geometry of a curve (its shape in $\mathbb{R}^3$) is intrinsically linked to the physics of motion. By differentiating the position vector $\mathbf{r}(t)$, we don't just get a rate—we get a direction. This allows us to predict where an object will be, how fast it is going, and how its path is curving at any given instant.

Formula / Concept Box

[!IMPORTANT] The velocity vector $\mathbf{v}(t)$ is always tangent to the path of the particle. The acceleration vector $\mathbf{a}(t)$, however, points toward the "inside" of the curve.

Hierarchical Outline

1. Foundations of Motion
   • $\mathbf{r}(t)$ as a Space Curve: Motion is a parameterization of a curve where $t$ is time.
   • Derivatives: Differentiating component-wise yields velocity and acceleration.
2. Kinematic Quantities
   • Speed vs. Velocity: Speed is the scalar magnitude; velocity is the directed vector.
   • Integration: Integrating $\mathbf{a}(t)$ with initial conditions $\mathbf{v}(0)$ and $\mathbf{r}(0)$ recovers the path.
3. Projectile Motion
   • Gravity: Constant acceleration $\mathbf{a}(t) = -g\mathbf{j}$ (where $g \approx 9.8 \, m/s^2$ or $32 , ft/s^2$).
   • Initial Velocity: Defined by angle $\theta$ and initial speed $v_0$.
4. Components of Acceleration
   • Tangential ($a_T$): Changes the speed.
   • Normal ($a_N$): Changes the direction (points toward center of curvature).

Visual Anchors

Kinematic Derivative Chain

Projectile Trajectory Breakdown

Definition-Example Pairs

• Definition: Speed Minimization The point in time where the magnitude of velocity is at its lowest. This often occurs at the peak of a trajectory in projectile motion.

  • Example: For \mathbf{r}(t) = \langle t$, $t^2 \rangle, velocity is \langle 1$, $2t \rangle. Speed is $\sqrt{1+4t^2}$. The minimum speed occurs at $t=0$, where $v(0) = 1$.

• Definition: Acceleration Proportionality When the acceleration vector is a scalar multiple of the position vector, often seen in circular or elliptical motion.

  • Example: For \mathbf{r}(t) = \langle \cos t$, $\sin t \rangle, \mathbf{a}(t) = \langle -\cos t$, $-\sin t \rangle. Here, $\mathbf{a}(t) = -1 \cdot \mathbf{r}(t)$.

Worked Examples

Example 1: Motion along a Parabola

Problem: A particle moves along $\mathbf{r}(t) = (t^2 - 4t)\mathbf{i} + (t-2)\mathbf{j}$. Find the velocity and acceleration at $t = 2$.

Step 1: Differentiate for Velocity $\mathbf{v}(t) = \mathbf{r}'(t) = \frac{d}{dt}(t^2 - 4t)\mathbf{i} + \frac{d}{dt}(t-2)\mathbf{j} = (2t-4)\mathbf{i} + 1\mathbf{j}$

Step 2: Differentiate for Acceleration $\mathbf{a}(t) = \mathbf{v}'(t) = \frac{d}{dt}(2t-4)\mathbf{i} + \frac{d}{dt}(1)\mathbf{j} = 2\mathbf{i} + 0\mathbf{j}$

Step 3: Evaluate at $t = 2$

• \mathbf{v}(2) =$ (2(2)-4)$\mathbf{i} + 1\mathbf{j} = 0\mathbf{i} + 1\mathbf{j} = \langle 0$, $1 \rangle
• \mathbf{a}(2) = 2\mathbf{i} = \langle 2$, $0 \rangle

Conclusion: At $t=2$, the particle is moving purely in the $y$-direction, but it is accelerating purely in the $x$-direction.

Checkpoint Questions

1. If $\mathbf{r}(t)$ is the position, what physical quantity does \|\mathbf{r}$'(t)\|$ represent?
2. True or False: The acceleration vector is always perpendicular to the velocity vector.
3. In projectile motion (ignoring air resistance), which component of the velocity vector remains constant?
4. If $\mathbf{a}(t) = \mathbf{0}$ for all $t$, what can you say about the path of the particle?

▶Click to see Answers

1. Speed.
2. False (only true if speed is constant).
3. The horizontal component ($x$-component), because gravity only acts vertically.
4. The particle moves in a straight line at a constant speed (linear motion).