A cat runs along a straight line with constant velocity of magnitude v. A dog chases the cat such that the velocity of dog is always directed towards the cat. The speed of dog is 'u' and always constant. At the instant both are separated by distance x and their velocities are mutually perpendicular, the magnitude of acceleration of dog is.

The diagram shows the dog's velocity just before and just after the moment in question, with a very small angle between them. Extending those arrows to the path of cat forms an isosceles triangle with height x and width v dt. (velocity of dog is always directed towards cat.) Magnitude of acceleration of dog at required moment

$=\frac{\left|{\overrightarrow{v}}_2-{\overrightarrow{v}}_1\right|}{dt} ......\left(\mathrm{i}\right)$

Now $\left|{\overrightarrow{v}}_2-{\overrightarrow{v}}_1\right|=ud\theta .....\left(\mathrm{ii}\right)$

$\text{and} d\theta =\frac{vdt}{x} .....\left(\mathrm{iii}\right)$

From equations (i), (ii) and (iii),

Required magnitude of acceleration $=\frac{uv}{x}$