Welcome to Resultant and Relative Velocity
Have you ever looked out of a train window at another train moving on the parallel track? If both trains are traveling at the same speed in the same direction, the other train appears to be completely standing still. If the other train is moving slightly faster, it looks like it is gently creeping ahead. This everyday experience is the heart of relative velocity.
In CCEA AS 2 Section B: Mechanics 2, we take this intuitive concept and apply vector mathematics to solve real-world problems involving moving ships, aircraft navigating through crosswinds, collision courses, and finding the shortest distance between two moving bodies.
Don't worry if vectors and bearings seem tricky at first. By breaking each scenario down into simple, step-by-step vector diagrams and equations, you will be able to tackle even the most demanding examination questions with confidence!
Quick Review of CCEA Exam Standards:
• Standard acceleration due to gravity is \(g = 9.8\text{ m s}^{-2}\) (unless stated otherwise).
• Non-exact numerical answers should always be given to 3 significant figures.
• Directions given as navigation bearings must always be written as 3-figure bearings measured clockwise from North (e.g., \(045^\circ\), \(090^\circ\), \(230^\circ\)).
1. Key Definitions: True vs. Relative Velocity
To master this chapter, we first need to distinguish between what is actually happening relative to the fixed ground and what appears to happen from the viewpoint of a moving observer.
1. True Velocity (\(\mathbf{v}\))
True velocity is the actual vector velocity of an object measured relative to a fixed frame of reference (usually the ground or Earth).
2. Relative Velocity (\(\mathbf{v}_{B/A}\))
Relative velocity is the velocity of an object \(B\) as observed from a moving object \(A\) (i.e., the velocity of \(B\) relative to \(A\)):
\(\mathbf{v}_{B/A} = \mathbf{v}_B - \mathbf{v}_A\)
Memory Aid: Notice the order of the letters: \(\mathbf{v}_{\text{Target}/\text{Observer}} = \mathbf{v}_{\text{Target}} - \mathbf{v}_{\text{Observer}}\). The letter that comes first is the first term in the subtraction.
3. Resultant Velocity in a Moving Medium
When an object travels through a moving medium—such as an aircraft flying through wind or a ferry crossing a flowing river—its true motion over the ground is the combined effect (resultant vector sum) of its engine velocity relative to the medium and the velocity of the medium itself:
\(\mathbf{v}_{\text{true}} = \mathbf{v}_{\text{rel}} + \mathbf{v}_{\text{medium}}\)
Here are two crucial navigation terms to keep clear in your mind:
• Heading (or Bearing Set): The direction in which the craft is pointed/steered by its engines. This determines the direction of \(\mathbf{v}_{\text{rel}}\) (e.g., airspeed or velocity in still water).
• Track (or Course Made Good): The actual path traveled over the ground. This is the direction of \(\mathbf{v}_{\text{true}}\).
Key Takeaway: If there is no wind or current, heading and track are identical. In the presence of wind or current, the craft must steer into the flow (heading) to achieve its desired path over the ground (track).
---2. Vector Techniques: Component Form and Triangles
Depending on how a problem is presented, you can use either Cartesian components (\(\mathbf{i}, \mathbf{j}\) notation) or geometric vector triangles (Sine/Cosine rules).
Method A: Cartesian Component Form (\(\mathbf{i}, \mathbf{j}\))
Let East be represented by unit vector \(\mathbf{i}\) and North by unit vector \(\mathbf{j}\).
If \(\mathbf{v}_A = x_A\mathbf{i} + y_A\mathbf{j}\) and \(\mathbf{v}_B = x_B\mathbf{i} + y_B\mathbf{j}\), then:
\(\mathbf{v}_{B/A} = \mathbf{v}_B - \mathbf{v}_A = (x_B - x_A)\mathbf{i} + (y_B - y_A)\mathbf{j}\)
From this relative velocity vector, you can find:
• Relative speed: \(|\mathbf{v}_{B/A}| = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}\)
• Direction: Calculate \(\tan\theta = \left|\frac{y_B - y_A}{x_B - x_A}\right|\) and sketch the vector components to find the correct 3-figure bearing from North.
Method B: Vector Triangles (Sine and Cosine Rules)
For navigation problems where speeds and bearings are given, sketching the vector addition triangle \(\mathbf{v}_{\text{true}} = \mathbf{v}_{\text{rel}} + \mathbf{v}_{\text{medium}}\) is often the fastest method.
• Sine Rule: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
• Cosine Rule: \(a^2 = b^2 + c^2 - 2bc\cos A\)
Example Walkthrough:
An aircraft has an airspeed of \(200\text{ km h}^{-1}\). A wind is blowing from the West towards the East at \(40\text{ km h}^{-1}\). The pilot wishes to fly due North.
1. The desired course/track is due North (\(\mathbf{v}_{\text{true}}\) points North).
2. The wind vector points due East (\(\mathbf{v}_{\text{wind}} = 40\mathbf{i}\)).
3. The pilot must head slightly North-West so that the eastward wind pushes the plane back onto a true North track.
4. Setting up the right-angled vector triangle: \(\sin\theta = \frac{40}{200} = 0.2 \implies \theta = 11.5^\circ\) West of North.
5. Therefore, the required heading is \(360^\circ - 11.5^\circ = 348.5^\circ \implies 349^\circ\) (to 3 s.f.).
6. The ground speed is \(|\mathbf{v}_{\text{true}}| = \sqrt{200^2 - 40^2} = \sqrt{38400} \approx 196\text{ km h}^{-1}\).
Key Takeaway: Always draw a clear vector triangle showing arrows flowing head-to-tail for addition: \(\mathbf{v}_{\text{rel}} + \mathbf{v}_{\text{medium}} = \mathbf{v}_{\text{true}}\).
---3. Relative Position, Interception, and Collision
When tracking two moving bodies, we can describe the position of body \(B\) relative to body \(A\) at any time \(t\).
The Relative Position Vector Equation
If object \(A\) starts at initial position \(\mathbf{r}_{A0}\) with constant velocity \(\mathbf{v}_A\), its position at time \(t\) is:
\(\mathbf{r}_A(t) = \mathbf{r}_{A0} + t\mathbf{v}_A\)
Similarly, for object \(B\):
\(\mathbf{r}_B(t) = \mathbf{r}_{B0} + t\mathbf{v}_B\)
The position of \(B\) relative to \(A\) at time \(t\) is:
\(\mathbf{r}_{B/A}(t) = \mathbf{r}_B(t) - \mathbf{r}_A(t) = (\mathbf{r}_{B0} - \mathbf{r}_{A0}) + t(\mathbf{v}_B - \mathbf{v}_A)\)
Condition for Interception or Collision
An interception or collision occurs if there is some positive time \(T > 0\) at which both objects are at the exact same location, meaning \(\mathbf{r}_{B/A}(T) = \mathbf{0}\):
\((\mathbf{r}_{B0} - \mathbf{r}_{A0}) + T(\mathbf{v}_B - \mathbf{v}_A) = \mathbf{0} \implies (\mathbf{r}_{B0} - \mathbf{r}_{A0}) = -T\mathbf{v}_{B/A}\)
This tells us two vital things for an interception:
1. The relative velocity vector \(\mathbf{v}_{B/A}\) must point directly along the line connecting their initial positions back towards \(A\).
2. The bearing of \(B\) as observed from \(A\) remains constant over time until impact.
Key Takeaway: Constant bearing between two moving craft means a collision will occur unless one of them changes speed or direction!
---4. Closest Approach (Shortest Distance)
When two objects do not collide, they will pass by each other. Finding their closest distance of approach and the time at which this occurs is a classic CCEA exam question. There are two robust methods you can use.
Method 1: Geometric Method (The Perpendicular Distance)
Imagine holding observer \(A\) fixed at the origin. From \(A\)'s viewpoint, body \(B\) starts at initial relative position \(\mathbf{r}_{B/A}(0) = \mathbf{r}_{B0} - \mathbf{r}_{A0}\) and travels in a straight line along the direction of the relative velocity vector \(\mathbf{v}_{B/A}\).
The shortest distance \(d_{\text{min}}\) from the fixed point \(A\) to this straight line path of relative motion is the perpendicular distance:
\(d_{\text{min}} = |\mathbf{r}_{B/A}(0)| \sin\alpha\)
where \(\alpha\) is the angle between the initial displacement vector \(\mathbf{r}_{B/A}(0)\) and the line of relative velocity \(\mathbf{v}_{B/A}\).
The time taken to reach the point of closest approach is:
\(t = \frac{|\mathbf{r}_{B/A}(0)| \cos\alpha}{|\mathbf{v}_{B/A}|}\)
Method 2: Vector / Dot Product / Calculus Method
The distance squared between \(B\) and \(A\) at any time \(t\) is \(D^2(t) = |\mathbf{r}_{B/A}(t)|^2\).
To find the minimum distance, we can use either of the following equivalent techniques:
1. Scalar Product Condition: At the instant of closest approach, the relative displacement vector is perpendicular to the relative velocity vector:
\(\mathbf{r}_{B/A}(t) \cdot \mathbf{v}_{B/A} = 0\)
2. Calculus: Write out the quadratic expression for \(D^2(t) = |\mathbf{r}_{B/A}(t)|^2\) in terms of \(t\), differentiate with respect to \(t\), and set \(\frac{d}{dt}(D^2) = 0\) to solve for \(t\). Substitute this \(t\) back into \(|\mathbf{r}_{B/A}(t)|\) to get \(d_{\text{min}}\).
Key Takeaway: Both methods give identical results! The dot product condition \(\mathbf{r}_{\text{rel}} \cdot \mathbf{v}_{\text{rel}} = 0\) is often the most direct algebraic method when working in \(\mathbf{i}, \mathbf{j}\) components.
---5. Common Pitfalls & Examiner Warnings
Be on your guard against these common errors reported in CCEA Mechanics examinations:
• Vector Subtraction Order: Confusing \(\mathbf{v}_{B/A} = \mathbf{v}_B - \mathbf{v}_A\) with \(\mathbf{v}_{A/B} = \mathbf{v}_A - \mathbf{v}_B\). Mixing these up reverses the direction of the vector by \(180^\circ\) and leads to incorrect bearings.
• Conflating Heading with Course/Track: Remember that an aircraft's nose points in the direction of the heading (\(\mathbf{v}_{\text{rel}}\)), but it actually moves along the track (\(\mathbf{v}_{\text{true}}\)). Do not put the airspeed along the ground track!
• Bearing Quadrant Errors: When calculating \(\tan\theta = \frac{v_y}{v_x}\), never write down \(\theta\) as your final bearing without drawing a sketch to check which quadrant the vector lies in. Navigation bearings must always be measured clockwise from North.
• Closest Approach Fallacy: Never assume that the closest approach occurs when one body reaches the starting point of the other. The closest approach is the perpendicular point along the relative path, found via \(\mathbf{r}_{\text{rel}} \cdot \mathbf{v}_{\text{rel}} = 0\) or geometry.
---Summary Checklist
Before sitting your AS 2 Mechanics exam, ensure you can confidently:
• State and apply \(\mathbf{v}_{B/A} = \mathbf{v}_B - \mathbf{v}_A\) and \(\mathbf{v}_{\text{true}} = \mathbf{v}_{\text{rel}} + \mathbf{v}_{\text{medium}}\).
• Construct accurate vector triangles using the Sine and Cosine rules for navigation problems.
• Form the time-dependent relative position vector \(\mathbf{r}_{B/A}(t) = \mathbf{r}_{B/A}(0) + t\mathbf{v}_{B/A}\).
• Test for interception using the condition of constant bearing or \(\mathbf{r}_{B/A}(T) = \mathbf{0}\).
• Calculate the time and distance of closest approach using \(\mathbf{r}_{B/A}(t) \cdot \mathbf{v}_{B/A} = 0\) or \(d_{\text{min}} = |\mathbf{r}_0|\sin\alpha\).
• Express all bearings as 3-figure values and final numerical answers to 3 significant figures.