# 38 - Rate of rotation of search light pointing to a ship

**Problem 38**

A ship, moving 8 mi/hr, sails north for 30 min, then turns east. If a searchlight at the point of departure follows the ship, how fast is the light rotating 2 hr after the start.

**Solution 38**

$\theta = \arctan \dfrac{8(t - 0.5)}{4}$

$\theta = \arctan 2(t - 0.5)$

$\dfrac{d\theta}{dt} = \dfrac{2}{1 + 4(t - 0.5)^2}$

after t = 2 hrs

$\dfrac{d\theta}{dt} = \dfrac{2}{1 + 4(2 - 0.5)^2}$

$\dfrac{d\theta}{dt} = \dfrac{2}{1 + 4(2.25)}$

$\dfrac{d\theta}{dt} = 0.2 \, \text{ rad/hr}$ *answer*