Trajectory collision calculation

In summary, trajectory collision calculation involves determining the point and time at which two moving objects will collide based on their trajectories, velocities, and positions. This process typically utilizes mathematical models and algorithms to predict interactions in various contexts, such as space missions, robotics, and gaming. Key factors include relative motion, acceleration, and potential obstacles, enabling accurate predictions that inform decision-making and safety measures.
  • #1
pejsek
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TL;DR Summary
Calculate α so that the red and blue objects meet using the given parameters
Hello,

I ask you for your aid in the solution of the following problem. Please see the attached illustration.

1692118566206.png


Two objects (red and blue) are moving in the vicinity of each other. The red object is moving along a closed circle and the blue object is moving along a line. Our objective is to make the two objects collide. We cannot touch the red object, however, we can change the direction of the movement of the blue object by changing the angle α. The goal is to calculate the angle α so that the two objects meet.

Here is what is known:
L ... distance between the two objects at time = 0
r ... radius of the trajectory of the red object
v_red ... magnitude of the tangential velocity of the red object
v_blue ... magnitude of the velocity of the blue object

Magnitudes of both velocities are constant (there is no acceleration except for the centripetal acceleration of the red object).

This problem comes from my idea of calculating the angle at which fighter planes have to fire their guns when shooting down an enemy aircraft in a turn.

Thank you very much indeed for your help,
pejsek
 
Last edited:
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  • #2
What's your approach to solving this problem? Do you have any ideas?
 
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  • #3
This is as far as I have got. I am either missing one more equation or the whole approach is wrong.

1692125326470.png
 
  • #4
Since A and B eventually has to be at the same position at the same time (a so-called kinematic intercept problem), perhaps you can formulate your problem in a way that facilitates a solution in that way?
 
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pejsek said:
This is as far as I have got. I am either missing one more equation or the whole approach is wrong.

View attachment 330573

Wow, that is mostly illegible for me. Please review the LaTeX Guide link at the lower left of the Edit box to learn how to post math equations here at PF. I will send you a separate Private Message (PM) with more information on using LaTeX.
 
  • #7
Have you tried solving an easier version of the problem first? For example, replacing the circular path with a straight path.
1692476665961.png

When facing something that initially looks too challenging it is often useful trying to solve similar easier things. It helps to build intuition and practice that can then be used in the original problem.
With the straight lines the system of equations will be significanlly easier and the solution will be unique since two straight lines only cross at one point. With the circular path the system of equations necessary will be more tedious to solve and you'll get more possible solutions since, normally, the straight line will be able to intersect the circle twice.
 

FAQ: Trajectory collision calculation

What is trajectory collision calculation?

Trajectory collision calculation is the process of determining if and when two or more moving objects will intersect or come into contact along their paths. This involves using mathematical models and algorithms to predict the future positions of the objects based on their current velocities, directions, and other relevant factors.

What are the key factors involved in trajectory collision calculations?

The key factors involved in trajectory collision calculations include the initial positions of the objects, their velocities (speed and direction), the influence of external forces (such as gravity or wind), and the time interval over which the prediction is made. Accurate calculations also require consideration of the objects' sizes and shapes to determine the exact point of collision.

Which mathematical methods are commonly used for trajectory collision calculations?

Common mathematical methods used for trajectory collision calculations include vector algebra, differential equations, and numerical integration techniques. Algorithms like the Newton-Raphson method, Runge-Kutta method, and Monte Carlo simulations are often employed to solve these equations and predict collisions accurately.

How are trajectory collision calculations applied in real-world scenarios?

Trajectory collision calculations are applied in various real-world scenarios, such as air traffic control to prevent airplane collisions, space missions to avoid satellite and debris collisions, robotics for path planning and obstacle avoidance, and automotive safety systems for collision detection and prevention. These calculations are critical for ensuring safety and efficiency in these applications.

What challenges are faced in trajectory collision calculations?

Challenges in trajectory collision calculations include accounting for uncertainties and variations in object movements, dealing with complex environments with multiple moving objects, and performing real-time calculations with high accuracy. Additionally, the computational complexity can be significant, requiring advanced algorithms and powerful computing resources to handle large-scale problems effectively.

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