## Statement

difficulty Two bodies, c1 and c2, begin to move from the same point at constant angular velocity, but in opposite directions, along a 30 m radius circumference. If the first one takes 20 seconds to complete a rotation and the second takes 60 seconds, calculate:

a) The time that they take to meet.
b) The angle and distance traveled by each one.

## Solution

Data

R = 30 m
T1 = 20 s
T2 = 60 s

Since their trajectory is a circumference and their angular velocities are constant we face a problem of uniform circular motion or u.c.m.

Question a)

Since we know the period of each one, we can calculate the angular velocities by means of the following equation:

$w=\frac{2\pi }{T}$

Therefore:

${w}_{1}=\frac{2\pi }{{T}_{1}}=\frac{2\pi }{20}=\frac{\pi }{10}rad\phantom{\rule{0ex}{0ex}}{w}_{2}=\frac{2\pi }{{T}_{2}}=\frac{2\pi }{60}=\frac{\pi }{30}rad$

Both bodies will meet each other before completing a rotation. In particular, when they meet the sum of their angular positions will be exactly 2π radians.

${\phi }_{1}+{\phi }_{2}=2\pi$

Therefore:

${\phi }_{1}=2\pi -{\phi }_{2}$

If we use the equation of position of ${\phi }_{1}$ , and substitute, we get:

Knowing that similarly ${\phi }_{2}={\omega }_{2}·t$, then:

Question b)

If we replace the time at which they meet (t = 15 s) and the angular velocity in the equation of position of ${\phi }_{1}$, we get that the angle traveled by c1, which is:

And that the angle traveled by c2 is:

To calculate now distance traveled (s) for each of the two bodies, simply apply the equation $s=\phi ·R$:

## Formulas worksheet

These are the main formulas that you must know to solve this exercise. If you are not clear about their meaning, we recommend you to check the theory in the corresponding sections. Furthermore, you will find in them, under the Formulas tab, the codes that will allow you to integrate these equations in external programs like Word or Mathematica.

Formulas
Related sections
$\phi ={\phi }_{0}+\omega ·t$
$s=\phi ·R$