You say goodbye to your friend at the intersection of two perpendicular roads. At time t=0 you drive off North at a (constant) speed v and your friend drives West at a (constant) speed . You badly want to know: how fast is the distance between you and your friend increasing at time t?

Explanation:

Rate of Change

When an object moves at constant speed v, the distance traveled at time t is

We know at time t=0 two friends are at the intersection of two perpendicular roads. One of them goes north at speed v and the other goes west at constant speed w (assumed). Since both directions are perpendicular, the distances make a right triangle. The vertical distance is

and the horizontal distance is

The distance between both friends is computed as the hypotenuse of the triangle

We need to find d', the rate of change of the distance between both friends.

Plugging in the above relations

Solving for d

Differentiating with respect to t

conservation of mass

mass before something happens = mass afterwards