Motion in a Plane and Relative Velocity

In Specialist Mathematics, motion in a plane is analyzed using vector functions. This allows for the simultaneous tracking of a particle’s horizontal and vertical components of motion relative to a fixed origin.

1. Vector Functions and Kinematics

A particle’s position at any time \(t\) is defined by a vector function \(\mathbf{r}(t)\). In two dimensions, this is expressed in terms of the unit vectors \(\mathbf{i}\) (horizontal) and \(\mathbf{j}\) (vertical).

Position, Velocity, and Acceleration

If the position vector is \(\mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j}\), then:

• Velocity (\(\mathbf{v}\) or \(\dot{\mathbf{r}}\)): The derivative of position with respect to time.
  \$\(\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = \dot{x}(t)\mathbf{i} + \dot{y}(t)\mathbf{j}\)\$
• Acceleration (\(\mathbf{a}\) or \(\ddot{\mathbf{r}}\)): The derivative of velocity with respect to time.
  \$\(\mathbf{a}(t) = \frac{d\mathbf{v}}{dt} = \ddot{x}(t)\mathbf{i} + \ddot{y}(t)\mathbf{j}\)\$
• Speed: The magnitude of the velocity vector.
  \$\(\text{Speed} = |\mathbf{v}(t)| = \sqrt{(\dot{x}(t))^2 + (\dot{y}(t))^2}\)\$

EXAM TIP: VCAA often requires the speed at a specific time. Ensure you differentiate the components correctly before substituting the value of \(t\) into the magnitude formula.

2. Path of a Particle (Cartesian Equations)

While the vector function \(\mathbf{r}(t)\) describes where and when a particle is at a point, the Cartesian equation describes the shape of the path by eliminating the parameter \(t\).

Common Conversions

COMMON MISTAKE: When sketching the path, pay close attention to the domain of \(t\). For example, if \(t \in [0, \pi]\), a circular path may only be a semi-circle. Always check if there are asymptotes (common in hyperbola paths involving \(\sec(t)\) or \(\tan(t)\)).

3. Relative Position and Velocity

Relative motion describes the motion of one object (A) as observed from another moving object (B).

Relative Position

The position of particle \(A\) relative to particle \(B\) is given by:
\$\(\mathbf{r}_{A/B} = \mathbf{r}_A - \mathbf{r}_B\)\$
If \(B\) is at the origin, then \(\mathbf{r}_{A/B} = \mathbf{r}_A\).

Relative Velocity

The velocity of particle \(A\) relative to particle \(B\) is the rate of change of the relative position:
\$\(\mathbf{v}_{A/B} = \mathbf{v}_A - \mathbf{v}_B\)\$

Constant Velocity Problems

If particles move with constant velocities \(\mathbf{u}\) and \(\mathbf{v}\), their positions at time \(t\) are:
\$\(\mathbf{r}_A(t) = \mathbf{r}_{A_0} + \mathbf{u}t\)\$
\$\(\mathbf{r}_B(t) = \mathbf{r}_{B_0} + \mathbf{v}t\)\$
where \(\mathbf{r}_{A_0}\) and \(\mathbf{r}_{B_0}\) are initial positions.

KEY TAKEAWAY: In relative velocity notation, \(\mathbf{v}_{A/B}\) is “Velocity of A minus Velocity of B”. Think of it as the velocity vector you would see if you were “sitting” on particle B.

4. Collision and Closest Approach

In 2D motion problems, two particles \(A\) and \(B\) are often analyzed to see if they collide or how close they pass to one another.

Collision

For a collision to occur, the particles must be at the same place at the same time:
\$\(\mathbf{r}_A(t) = \mathbf{r}_B(t)\)\$
This requires solving the simultaneous equations:
1. \(x_A(t) = x_B(t)\)
2. \(y_A(t) = y_B(t)\)
Note: Both equations must yield the same value for \(t\).

Closest Approach (Minimum Distance)

The distance \(D\) between two particles at time \(t\) is the magnitude of the relative position vector:
\$\(D(t) = |\mathbf{r}_{A/B}(t)| = |\mathbf{r}_A(t) - \mathbf{r}_B(t)|\)\$

To find the minimum distance:
1. Define the squared distance function \(f(t) = [D(t)]^2\). (Minimizing \(D^2\) is easier than minimizing \(D\)).
2. Solve \(f'(t) = 0\) for \(t\).
3. Alternatively, use the geometric property: at the time of closest approach, the relative position vector is perpendicular to the relative velocity vector.
\$\(\mathbf{r}_{A/B}(t) \cdot \mathbf{v}_{A/B}(t) = 0\)\$

VCAA FOCUS: Problems involving a “police boat” intercepting a “motorboat” are common. You may be asked to find a specific velocity vector (or a scalar multiplier \(u\)) that ensures an interception. This is solved by setting \(\mathbf{r}_A(t) = \mathbf{r}_B(t)\) and solving for both \(t\) and the unknown velocity component.

5. Summary of Key Formulas

• Displacement: \(\Delta \mathbf{r} = \int_{t_1}^{t_2} \mathbf{v}(t) \, dt\)
• Distance Travelled (along the path): \(s = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\)
• Unit vector in direction of motion: \(\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}\)
• Perpendicular Vectors: If \(\mathbf{a} = x\mathbf{i} + y\mathbf{j}\), then a perpendicular vector is \(\mathbf{b} = y\mathbf{i} - x\mathbf{j}\) (or \(-y\mathbf{i} + x\mathbf{j}\)).

REMEMBER: Speed is a scalar (\(|\mathbf{v}|\)), while velocity is a vector (\(\mathbf{v}\)). If an exam question asks for velocity, provide the answer in \(\mathbf{i}, \mathbf{j}\) form or as a magnitude and direction (bearing). If it asks for speed, provide a magnitude only.