Question

A Saharan silver ant (Cataglyphis bombycina) can travel at 885 mm/s. (This speed is measured relative to the surface the ant is running on.) The ant is running across a treadmill which is set to 8 kilometers per hour. If the ant runs perpendicular to the treadmill’s motion, what is its speed? If it runs against the treadmill’s motion (as a person would, if they were running on the treadmill) what is the ant’s velocity? (Be sure to define a coordinate system.)

Answer 1

885 mm/s is 0.885 m/s.

8 km/h is 8*1000/3600 = 2.22 m/s.

If the ant runs perpendicular to the treadmill’s motion, its speed equals the magnitude of the vector sum of velocity vectors of the treadmill and ant:

$$v_\perp=\sqrt{0.885^2+2.22^2}=2.39\text{ m/s}.$$

This speed is relative to the ground.

If it runs against the treadmill’s motion, relative to the ground the ant's sped will be

$$v_{\downarrow\uparrow}=2.22-0.885=1.34\text{ m/s}.$$