An object moves towards another object, which moves in straight line

[visharad]

A cat and a mouse see each other at the same time. The distance between them is 40 meters. The mouse moves in a straight line - perpendicularly to the line joining the two. The cat always moves towards the mouse. Each moves at 20 m/s. What is the closest distance the cat can reach the mouse?

I am not really sure how to work this out. I could only figure out the following:-
The cat moves along a curve. At any moment, the line joining the cat and the mouse is a tangent to the curve.

 

[AlephZero]

Think about the mouse's velocity relative to the cat.

Is the distance between cat and mouse increasing or decreasing?

Does this change in distance depend on the current distance between the cat and mouse, or not?

 

[visharad]

Initially the distance decreases. But it is possible that later it increases.
So the change in distance depends on their current positions.
But I really do not know how this helps.

 

[visharad]

If I could somehow find the relative position between the cat and the mouse in terms of time, then the problem will become easy. But I do not know how to find the relative position.
In know their relative initial position and initial velocity but I do not know how the relative acceleration changes with time.

 

[AlephZero]

Ahh... it depends how you read the question. I read it as meaning "the mouse always moves perpendicular to the line joining it to the cat." In that case, both the cat and the mouse run in a curve. From the relative velocities, the distance ALWAYS decreases, so the cat catches the mouse.

But it actually says "the mouse moves in a straight line" that it initially at right angles to the cat. In that case, you can write down an equation for the motion of the mouse, and then set up a differential equation for the motion of the cat. See here for how to solve the equations:
http://mathworld.wolfram.com/PursuitCurve.html

 

[visharad]

The mouse moves in a straight line. Suppose initially the mouse is at A and the cat is at B, then the mouse moves perpendicular to the line AB. The cat always moves towards the mouse.
Both move at 20 m/s. Their initial separation = 40 m
Seeing the equations in that page.
Thanks.