Welcome to Constant Velocity in Two Dimensions!
Ever wondered how air traffic controllers keep hundreds of planes moving without them ever bumping into each other? Or how a captain navigates a ship across a windy ocean to hit a tiny port thousands of miles away? The secret lies in vectors and constant velocity.
In this chapter of Further Mechanics (FM1), we are moving away from simple "left and right" motion and stepping into a 2D world. We will learn how to describe exactly where an object is and where it's going using coordinates and vectors. Don't worry if vectors felt a bit strange in your previous math units—we will break everything down step-by-step!
1. The Basics: Displacement, Speed, and Velocity
Before we dive into the math, let's make sure we are clear on our terms. In Mechanics, we care about the difference between "how far you went" and "where you ended up."
- Displacement (\(\mathbf{r}\) or \(\mathbf{s}\)): This is a vector. It tells us the straight-line distance and the direction from a starting point (the origin).
- Velocity (\(\mathbf{v}\)): This is also a vector. It tells us how fast something is moving and in what direction.
- Speed: This is a scalar. It is just the magnitude (the "size") of the velocity. We find this using Pythagoras’ Theorem on the velocity vector.
How to write vectors
In your exam, you can see vectors written in two ways. Both mean the exact same thing!
- Unit Vector Form: \(a\mathbf{i} + b\mathbf{j}\) (where \(\mathbf{i}\) is 1 unit Right and \(\mathbf{j}\) is 1 unit Up).
- Column Vector Form: \(\begin{pmatrix} a \\ b \end{pmatrix}\)
Quick Review: To find the speed from a velocity vector \(\mathbf{v} = x\mathbf{i} + y\mathbf{j}\), use:
\( \text{Speed} = |\mathbf{v}| = \sqrt{x^2 + y^2} \)
2. The Fundamental Position Equation
If an object starts at a certain position and moves at a constant velocity, we can predict exactly where it will be at any time \(t\).
The formula is:
\( \mathbf{r} = \mathbf{r}_0 + \mathbf{v}t \)
Where:
\(\mathbf{r}\) = Position at time \(t\)
\(\mathbf{r}_0\) = Initial position (where it started at \(t = 0\))
\(\mathbf{v}\) = Constant velocity vector
\(t\) = Time elapsed
Example: A robot starts at position \(2\mathbf{i} + 3\mathbf{j}\) and moves with velocity \(4\mathbf{i} - 1\mathbf{j}\) for 3 seconds. Where is it?
\( \mathbf{r} = (2\mathbf{i} + 3\mathbf{j}) + (4\mathbf{i} - 1\mathbf{j}) \times 3 \)
\( \mathbf{r} = (2 + 12)\mathbf{i} + (3 - 3)\mathbf{j} = 14\mathbf{i} \)
Key Takeaway: The position vector is just the starting point plus the "velocity jump" multiplied by time.
3. Resultant Velocities
Sometimes, an object has two velocities acting on it at once. Think of a boat trying to cross a river. The boat has its own engine speed, but the river current is also pushing it.
To find the resultant velocity (the actual path the boat takes), we simply add the vectors together.
Methods to solve:
1. Vector Addition: If \(\mathbf{v}_{boat} = \begin{pmatrix} 5 \\ 0 \end{pmatrix}\) and \(\mathbf{v}_{river} = \begin{pmatrix} 0 \\ 2 \end{pmatrix}\), the resultant is \(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\).
2. Vector Triangles: You can draw the vectors "tip-to-tail." This often creates a right-angled triangle where you can use Sine, Cosine, or Pythagoras.
Did you know? Pilots have to point their planes slightly "into the wind" so that the resultant velocity keeps them on a straight track to the airport. This is called "crab angle"!
4. Relative Velocity
Relative velocity is how fast something appears to be moving from the perspective of another moving object.
The Golden Rule: To find the velocity of A relative to B (written as \(\mathbf{v}_{A|B}\) or \({}_B\mathbf{v}_A\)), you subtract the velocity of the observer (B) from the velocity of the object (A).
\( \mathbf{v}_{A|B} = \mathbf{v}_A - \mathbf{v}_B \)
Analogy: If you are in a car doing 60 mph and a car passes you at 70 mph, it only looks like they are going 10 mph to you. That 10 mph is the relative velocity.
Memory Aid: "A relative to B" = A minus B.
5. Interception and Closest Approach
This is often the most challenging part of the chapter, but we can break it down into a clear process.
Interception (Collision)
Two objects, A and B, will collide if they are at the same position at the same time.
To solve: Set \(\mathbf{r}_A = \mathbf{r}_B\) and solve for \(t\). If you get the same value of \(t\) for both the \(\mathbf{i}\) and \(\mathbf{j}\) components, they collide!
Closest Approach
If two objects don't collide, they will still have a point where they are nearest to each other. To find this distance:
- Find the relative position vector: \(\mathbf{r}_{A|B} = \mathbf{r}_A - \mathbf{r}_B\).
- This will give you an expression in terms of \(t\), such as \(\begin{pmatrix} 2t-4 \\ t+3 \end{pmatrix}\).
- Find the distance squared (\(d^2\)) using Pythagoras: \(d^2 = (2t-4)^2 + (t+3)^2\).
- To find the minimum distance, you have two choices:
- Calculus: Differentiate the expression (\(\frac{d(d^2)}{dt}\)), set it to zero, and solve for \(t\).
- Completing the Square: Rewrite the quadratic to find the minimum value.
- Plug that \(t\) back into your distance formula to find the actual minimum distance.
Common Mistake: Students often forget that \(d^2\) is being minimized. Don't forget to take the square root at the very end to get the actual distance \(d\)!
Quick Review Box
1. Position: \(\mathbf{r} = \mathbf{r}_0 + \mathbf{v}t\)
2. Speed: Magnitude of velocity \(\sqrt{x^2 + y^2}\)
3. Relative Velocity: \(\mathbf{v}_{A|B} = \mathbf{v}_A - \mathbf{v}_B\)
4. Relative Position: \(\mathbf{r}_{A|B} = \mathbf{r}_A - \mathbf{r}_B\)
5. Closest Approach: Minimize the distance squared \(d^2\) between the two objects.
Don't worry if the relative velocity part feels tricky at first. Just remember that you are "freezing" one object and looking at the world through its eyes! Practice drawing the vectors, and the patterns will start to make sense.