Kinematics and Dynamics

Kinematics is the study of the motion of points, objects, and groups of objects without considering the causes of motion (forces). In VCE Specialist Mathematics, this involves the application of calculus and vectors to describe motion in one and two dimensions.

1. Rectilinear Motion (Motion in a Straight Line)

Rectilinear motion involves an object moving along a straight line (usually the $x$-axis). Its position, velocity, and acceleration are functions of time $t$.

Fundamental Relationships

Given position $x(t)$:
* Displacement: The change in position, $\Delta x = x(t_2) - x(t_1)$.
* Velocity ($v$): The rate of change of displacement with respect to time.
$$v = \frac{dx}{dt}$$
* Acceleration ($a$): The rate of change of velocity with respect to time.
$$a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$$

Integration in Kinematics

To move “up” from acceleration to position, we integrate with respect to time:
* $v(t) = \int a(t) \, dt$
* $x(t) = \int v(t) \, dt$

Alternative Forms of Acceleration

When acceleration is given as a function of velocity $v$ or displacement $x$, we use alternative chain rule forms:
1. $a = \frac{dv}{dt}$ (used when $a$ is a function of $t$)
2. $a = v \frac{dv}{dx}$ (used when $a$ is a function of $x$)
3. $a = \frac{d}{dx} \left( \frac{1}{2}v^2 \right)$ (useful for energy-related problems or integrating with respect to $x$)

EXAM TIP: When you see acceleration expressed in terms of displacement ($x$), almost always use $a = v \frac{dv}{dx}$ or $a = \frac{d}{dx}(\frac{1}{2}v^2)$. This allows you to separate variables and integrate: $\int v \, dv = \int a(x) \, dx$.

2. Distance vs Displacement

• Displacement: The vector quantity representing the change in position.
  $$\text{Displacement} = \int_{t_1}^{t_2} v(t) \, dt$$
• Distance: The total path length traveled (a scalar).
  $$\text{Distance} = \int_{t_1}^{t_2} |v(t)| \, dt$$

COMMON MISTAKE: Students often confuse distance and displacement. If a particle moves forward and then backward, the displacement might be zero, but the distance is the sum of the absolute values of the movements. Always check if the velocity $v(t)$ crosses the $t$-axis (changes direction) within the given interval.

3. Vector Kinematics in Two Dimensions

In 2D motion, we use position vectors $\mathbf{r}(t)$ relative to an origin $O$.

Vector Definitions

• Position Vector: $\mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j}$
• Velocity Vector: $\mathbf{v}(t) = \dot{x}(t)\mathbf{i} + \dot{y}(t)\mathbf{j}$
• Acceleration Vector: $\mathbf{a}(t) = \ddot{x}(t)\mathbf{i} + \ddot{y}(t)\mathbf{j}$

Key Scalar Quantities

• Speed: The magnitude of the velocity vector.
  $$\text{Speed} = |\mathbf{v}(t)| = \sqrt{(\dot{x})^2 + (\dot{y})^2}$$
• Distance Traveled (Arc Length):
  $$s = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$

KEY TAKEAWAY: In vector kinematics, the $\mathbf{i}$ and $\mathbf{j}$ components are independent. You can differentiate or integrate them separately to find the components of velocity and acceleration.

4. Projectile Motion

Projectile motion is a specific case of 2D motion under constant vertical acceleration (gravity) and zero horizontal acceleration (ignoring air resistance).

Standard Equations

Assuming a projectile is launched from $(0,0)$ with initial speed $V$ at an angle $\theta$ to the horizontal:

Cartesian Equation of the Path (Trajectory)

By eliminating $t$ from the position equations ($t = \frac{x}{V\cos\theta}$), we get:
$$y = x\tan(\theta) - \frac{gx^2}{2V^2\cos^2(\theta)}$$
Using the identity $\sec^2(\theta) = 1 + \tan^2(\theta)$:
$$y = x\tan(\theta) - \frac{gx^2}{2V^2}(1 + \tan^2(\theta))$$

Key Projectile Metrics (for launch from ground level)

• Time of Flight: Set $y=0 \implies T = \frac{2V\sin(\theta)}{g}$
• Maximum Height: Set $v_y=0 \implies H = \frac{V^2\sin^2(\theta)}{2g}$
• Horizontal Range: $R = \frac{V^2\sin(2\theta)}{g}$

VCAA FOCUS: VCAA often asks questions where the launch point is not the origin or the landing surface is inclined. Do not rely solely on the Range/Height formulas; be prepared to derive results from the basic $x(t)$ and $y(t)$ equations.

5. Circular Motion (Kinematics)

When a particle moves in a circle of radius $r$ with constant angular speed $\omega = \frac{d\theta}{dt}$.

Vector Representation

The position vector can be defined as:
$$\mathbf{r}(t) = r\cos(\omega t)\mathbf{i} + r\sin(\omega t)\mathbf{j}$$

Velocity and Acceleration

• Velocity: $\mathbf{v}(t) = -r\omega\sin(\omega t)\mathbf{i} + r\omega\cos(\omega t)\mathbf{j}$
  • Speed is constant: $|\mathbf{v}| = r\omega$
• Acceleration: $\mathbf{a}(t) = -r\omega^2\cos(\omega t)\mathbf{i} - r\omega^2\sin(\omega t)\mathbf{j}$
  • This can be simplified to: $\mathbf{a}(t) = -\omega^2 \mathbf{r}(t)$

Centripetal Acceleration

The acceleration vector points directly toward the center of the circle (opposite to the position vector).
* Magnitude: $a = r\omega^2 = \frac{v^2}{r}$

REMEMBER: In uniform circular motion, even though the speed is constant, the velocity is not constant because its direction is constantly changing. Therefore, there is always a non-zero acceleration directed toward the center.

6. Summary Table: Calculus Relationships

STUDY HINT: Practice converting between parametric vector equations $\mathbf{r}(t)$ and Cartesian equations $y=f(x)$. This is a common requirement in “Path of the Particle” questions. Use trigonometric identities (like $\sin^2\theta + \cos^2\theta = 1$) to eliminate the parameter $t$.