A Chase





The applet shows a chase. You can see the blue ball's velocity being always directed towards the position of the red ball.

It should be interesting to see the result of making the velocities of the two equal. The applet is initially set to start with equal velocities for the two balls. Change both the velocities while keeping them equal and see what happens.

When the speeds of the two balls are equal, the distance between the balls tends to half the initial distance and this would be attained after infinite time. The applet does not of course wait for that to happen, it is reset to the initial conditions after a while. You should be able to notice that the distance quickly falls to a value close to half the initial distance. ( You may also notice some inaccuracy when y-distance becomes very small, the distance is shown as equal to x-distance, which can be understood as an equality within the limits of accuracy imposed.)

With x axis to the right and y axis downward and theta denoting the angle made by blue ball's velocity with x-axis, the following equations could be written.

This equation is for the x displacement of the red particle.

This equation is for the x displacement of the blue particle.

Since the initial x co-ordiantes of the two are same, the diifference between the two x displacements would be equal to the x distance between the two.

This x distance is shown as a white line in the applet.

Choosing the line joining the two balls as a reference and splitting the velocity of the red ball in to components( notice that the angle between the red ball's velocity and line joining the two balls is same as the angle between blue ball's velocity and the x-axis), the equation for the change in the length of the line joining the two balls is

This distance is shown as a yellow line in the applet.

The green line gives the y-distance between the two balls. The equation for this is not necessary for the analysis.

In case when the velocities are equal the right hand sides of the last two equations add up to zero. This would mean that the sum of the changes in the x-distance represented by the length of the white line and the distance- represented by the yellow line must be zero. In the applet the x-distance increases by the same amount as the distance decreases. Since initially the x-distance is zero this sum is simply equal to the initial distance, which is 300. After a long time the x- distance should approach the distance. And since finally ( at t= infinity) the x-distance and distance should be equal ( because y-distance is zero), that distance should be half of the initial distance. In this case that is 150.