Collision and Interception: Relative Velocity Vectors

1. The Mathematical Condition for Collision

The most direct way to derive the condition for collision is to start from our result for minimum separation from the previous lesson:
$d_{\min }=\frac{|{\vec{r}}_{B/A}\times {\vec{v}}_{B/A}|}{|{\vec{v}}_{B/A}|}$
A collision or interception occurs if and only if the minimum separation distance is zero. Setting $d_{\min }=0$ in the equation above immediately leads to the conclusion that the numerator must be zero:
$|{\vec{r}}_{B/A}(0)\times {\vec{v}}_{B/A}|=0$
Here, ${\vec{r}}_{B/A}(0)$ is the initial relative position vector, and ${\vec{v}}_{B/A}$ is the constant relative velocity.

The cross product of two non-zero vectors is zero if and only if they are parallel (or anti-parallel). Therefore, the rigorous condition for a collision is:
${\vec{r}}_{B/A}(0)\times {\vec{v}}_{B/A}=\vec{0}$
This means that for a collision to occur, the relative velocity vector must be collinear with the initial relative position vector.

Let's visualize what this means. In the reference frame of object A, A is stationary at the origin. For object B to collide with A, its path must be a straight line aimed directly at the origin. This implies that its velocity vector in this frame, ${\vec{v}}_{B/A}$, must point exactly along its initial position vector relative to the origin, ${\vec{r}}_{B/A}(0)$, but in the opposite direction.

2. Solving Interception Problems

This condition is the key to solving interception problems, where you typically need to find the direction an object must travel to intercept another.

To see how this condition simplifies in practice, let's consider a common setup. Suppose at $t=0$, object A is at the origin and object B is at $(d,0)$.

• Initial relative position: ${\vec{r}}_{A/B}(0)={\vec{r}}_A(0)-{\vec{r}}_B(0)=\vec{0}-d\hat{i}=-d\hat{i}$.
• Relative velocity: ${\vec{v}}_{A/B}={\vec{v}}_A-{\vec{v}}_B=(v_{Ax}-v_{Bx})\hat{i}+(v_{Ay}-v_{By})\hat{j}$.

Now, let's apply the collision condition ${\vec{r}}_{A/B}(0)\times {\vec{v}}_{A/B}=\vec{0}$:
$(-d\hat{i})\times [(v_{Ax}-v_{Bx})\hat{i}+(v_{Ay}-v_{By})\hat{j}]=\vec{0}$
Using the distributive property of the cross product and the facts that $\hat{i}\times \hat{i}=\vec{0}$ and $\hat{i}\times \hat{j}=\hat{k}$:
$-d(v_{Ay}-v_{By})(\hat{i}\times \hat{j})=\vec{0}$
$-d(v_{Ay}-v_{By})\hat{k}=\vec{0}$
Since $d\ne 0$, this forces the term in the parenthesis to be zero:
$v_{Ay}-v_{By}=0⟹v_{Ay}=v_{By}$

This reveals a powerful strategy: For two objects to collide, the components of their velocities perpendicular to the line initially joining them must be equal. This ensures that their relative velocity has no component perpendicular to this line, meaning the relative motion occurs purely along the line connecting them.

The following resource provides a clear solution strategy based on this principle and includes worked examples.

3. Time to Collision

Once we've established the condition for a collision course, calculating the time it takes for the interception to happen is straightforward. In the relative frame, one object moves directly towards the other (which is stationary). The problem is reduced to 1D motion.

• The distance to be covered is the initial separation, $|{\vec{r}}_{B/A}(0)|$.
• The speed of approach is the magnitude of the relative velocity, $|{\vec{v}}_{B/A}|$.

Therefore, the time to collision is:
$t_{\text{collision}}=\frac{\text{Initial Separation Distance}}{\text{Magnitude of Relative Velocity}}=\frac{|{\vec{r}}_{B/A}(0)|}{|{\vec{v}}_{B/A}|}$

This is a direct application of time = distance / speed in the relative frame.

4. Practice Problems

To solidify your understanding, it is crucial to solve a variety of interception problems. The resource you read earlier provides an excellent set of exercises.

Conclusion

Today we have rigorously derived the condition for collision, showing it to be a special case of the minimum distance problem where the minimum distance is zero. This led us to the fundamental requirement that the relative velocity vector must be collinear with the initial relative position vector.

Key Takeaways:

• The formal condition for a collision between objects A and B is ${\vec{r}}_{A/B}(0)\times {\vec{v}}_{A/B}=\vec{0}$.
• In practice, this simplifies to a powerful problem-solving technique: the components of the objects' velocities perpendicular to the line initially joining them must be equal.
• Once the collision course is determined, the time to collision is found by dividing the initial separation distance by the magnitude of the relative velocity: $t=|{\vec{r}}_{\text{rel}}|/|{\vec{v}}_{\text{rel}}|$.

Preview of the Next Module:
We have now concluded our detailed study of relative motion. In the next module, we will shift our focus to a new type of motion: circular motion. We will begin by defining angular kinematic quantities (angular position, velocity, and acceleration) and deriving their relationships using calculus, laying the foundation for analyzing objects moving in circles.