Understanding the Shortest Distance Between Two Moving Vehicles Perpendicular to Each Other

The shortest distance between two moving vehicles that are moving in perpendicular directions can be determined using basic kinematics and vector principles. Let's explore the mathematical framework and the significance of this concept in real-world scenarios.

Setup and Initial Positions

Let there be two vehicles on a coordinate system, where Vehicle A moves along the x-axis and Vehicle B moves along the y-axis. Assume that at time ( t 0 ), both vehicles start from the same point, the origin (0,0). Vehicle A moves at a speed of ( v_A ) (m/s) in the east-to-west direction, while Vehicle B moves at a speed of ( v_B ) (m/s) in the south-to-north direction.

Position as a Function of Time

Position of Vehicle A

The position of Vehicle A at any time ( t ) is given by:

( x_A v_A t )

( y_A 0 )

Position of Vehicle B

The position of Vehicle B at any time ( t ) is given by:

( x_B 0 )

( y_B v_B t )

Distance Between the Two Vehicles

The distance ( d ) between the two vehicles at any time ( t ) can be calculated using the distance formula:

( d(t) sqrt{(x_A - x_B)^2 (y_A - y_B)^2} )

Substituting the positions of the vehicles, we get:

( d(t) sqrt{(v_A t - 0)^2 (0 - v_B t)^2} )

This simplifies to:

( d(t) sqrt{v_A^2 t^2 v_B^2 t^2} )

( d(t) sqrt{(v_A^2 v_B^2) t^2} )

( d(t) t sqrt{v_A^2 v_B^2} )

Conclusion and Analysis

Since the distance ( d(t) ) is directly proportional to time ( t ), the distance between the two vehicles will increase over time as they move in perpendicular directions. The shortest distance occurs at the time ( t 0 ), which is when the vehicles start at the origin. Therefore, the initial distance between the two vehicles is zero.

If a specific time ( t ) is required, the distance can be calculated using the formula:

( d(t) t sqrt{v_A^2 v_B^2} )

Real-World Application

In practical scenarios, this concept is vital for traffic management, collision avoidance systems, and navigation applications. The real-time tracking of vehicles can help in predicting the closest approach and thus, aid in safety measures.

Additional Insights

To further illustrate, consider the case where the positions of the vehicles are governed by the equations:

Distance from Point 1 to Intersection: ( x )

Distance from Point 2 to Intersection: ( y )

The time at which the distance between the vehicles is minimum is given by:

( t_{text{min}} frac{x v_1 y v_2}{v_1^2 v_2^2} )

This formula can help in optimizing the routing and scheduling of transportation.