Question

Two people start at the same place and walk around a circular lake in opposite directions. one walks with an angular speed of 1.80 10-3 rad/s, while the other has an angular speed of 3.70 10-3 rad/s. how long will it be before they meet?

Answer 1

The angle that corresponds to a complete revolution around the lake is
`2 \pi`.

Taking the direction of the first person as positive direction, his angular position around the lake is

`\theta_1 (t) = \omega_1 t`
where
`\omega_1 = 1.80 \cdot 10^(-3) rad/s` is the angular speed of the first person.

The second person is going in the opposite direction, so we can write his angular position around the lake as

`\theta_2 (t) = 2 \pi - \omega_2 t`
where
`\omega_2 = 3.70 \cdot 10^(-3)rad/s` is his angular speed, and
`2 \pi` is the angle that corresponds to one complete revolution around the lake.

The two people meet when their angular position is the same:

`\theta_1 (t) = \theta _2 (t)`

`\omega_1 t = 2 \pi - \omega_2 t`
from which we find the time t after which they meet again:

`t= (2\pi)/(\omega_1 + \omega_2) = (2 \pi)/(1.80 \cdot 10^(-3) rad/s + 3.7 \cdot 10^(-3) rad/s)=1142 s`