AP Calculus BC 9.6 Solving Motion Problems Using Parametric and Vector- Valued Functions Study Notes - New Syllabus

Solving Motion Problems Using Parametric and Vector-Valued Functions

Motion in a plane can be described using either parametric equations or vector-valued functions. These methods allow us to model movement in two or three dimensions and to compute position, velocity, and acceleration as functions of time.

Position Functions

• In parametric form: $ x = f(t), \quad y = g(t) $ Here, \(t\) represents time, \(x\) and \(y\) describe the coordinates of the moving object.
• In vector-valued form: $ \mathbf{r}(t) = \langle f(t), g(t) \rangle \quad \text{(2D)}, \quad \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \quad \text{(3D)} $ The vector \(\mathbf{r}(t)\) gives the position of the particle at time \(t\).

Velocity Functions

• Velocity is the time derivative of position: $ \mathbf{v}(t) = \mathbf{r}'(t) = \left\langle \dfrac{dx}{dt}, \dfrac{dy}{dt} \right\rangle $
• The magnitude of velocity is called speed: $ \text{Speed} = |\mathbf{v}(t)| = \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} $
• Velocity gives both the speed and the direction of motion.

Acceleration Functions

• Acceleration is the time derivative of velocity: $ \mathbf{a}(t) = \mathbf{v}'(t) = \left\langle \dfrac{d^2x}{dt^2}, \dfrac{d^2y}{dt^2} \right\rangle $
• Acceleration describes how the velocity changes over time.

Speed, Distance, and Displacement

• Displacement Vector: $ \mathbf{D} = \mathbf{r}(t_2) – \mathbf{r}(t_1) $ Represents the change in position from \(t_1\) to \(t_2\).
• Total Distance Traveled: $ \text{Distance} = \int_{t_1}^{t_2} |\mathbf{v}(t)| \, dt $ This measures the total path length regardless of direction.
• Speed at a Given Time: $ |\mathbf{v}(t)| = \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} $

Motion in Three Dimensions

• In 3D, position is given by: $ \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle $
• Velocity and acceleration are: $ \mathbf{v}(t) = \langle x'(t), y'(t), z'(t) \rangle, \quad \mathbf{a}(t) = \langle x”(t), y”(t), z”(t) \rangle $
• All formulas for speed, displacement, and distance apply similarly in three dimensions.

Tangential and Normal Components of Acceleration

• Acceleration can be decomposed into two perpendicular components: $ a_T = \frac{d}{dt}|\mathbf{v}(t)| \quad \text{(tangential acceleration)} $ $ a_N = \sqrt{|\mathbf{a}(t)|^2 – a_T^2} \quad \text{(normal acceleration)} $
• \(a_T\) changes the speed of the particle, \(a_N\) changes its direction.

 Unit Tangent and Normal Vectors

• Unit Tangent Vector: $ \mathbf{T}(t) = \frac{\mathbf{v}(t)}{|\mathbf{v}(t)|} $ Represents the direction of motion.
• Unit Normal Vector: $ \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|} $ Points towards the center of curvature of the path.

Practical Uses

• Modeling projectile motion
• Tracking satellites or spacecraft trajectories
• Simulating car motion on curved roads
• Analyzing roller coaster paths
• Studying fluid particle motion in physics