# Related rates?

## Main Question or Discussion Point

Hi,

Is this a question on related rates? If so, can anyone help me do this question? I have no clue on even how to start it out.

A clock has an hour hand 4cm long and a minute hand 8cm long. How fast is the distance between the tips of the hands changing at 2:00?

arildno
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1. Describe the position vectors of the tips as follows:
$$\vec{r}_{hour}(t)=4(\cos(\omega_{h}t+\theta_{h,2})\vec{i}+\sin(\omega_{h}t+\theta_{h,2})\vec{j})$$
$$\vec{r}_{min}(t)=8(\cos(\omega_{m}t+\theta_{m,2})\vec{i}+\sin(\omega_{m}t+\theta_{m,2})\vec{j})$$
Here:
$$\omega_{h},\omega_{m}$$ are the respective angular velocities (you should be able to figure them out by yourself).
$$\theta_{h,2},\theta_{m,2}$$ are the respective angles made to the positive x-axis (that is, 3 o'clock) at time 2.00

2. The distance vector $$\vec{d}(t)$$ fullfills:
$$\vec{d}(t)=\vec{r}_{hour}(t)-\vec{r}_{min}(t)$$
The distance, d(t) fullfills:
$$d(t)=||\vec{d}(t)||$$ that is, the LENGTH of the distance vector.
3. The separation velocity is therefore given as:
$$\frac{dd}{dt}$$