Question

Two cars are moving along a circular track of radius 40 m. The two cars were moving with constant angular speeds of 0.13 rad/s and 0.18 rad/s respectively. If they start from opposite sides of the track and move in opposite directions, how long will it take them to meet?

Draw a diagram to help.

Answer 1

Do you know how many radians are in a complete circle?

Answer 2

2 pi radians make up a circle, right?

Answer 3

Relate angles to angular speed, then equate both angles.

Answer should be around 8 seconds.

Answer 4

How would I begin to approach this problem? We haven't learned the appropriate equations for it.

Answer 5

You're correct! There are 2 pi radians in a circle.

We are given the angular velocity of both the cars.

Let's practice with a simpler problem only involving one car. I'll use degrees in this problem, as sometimes they are more familiar.

Let's say a car drives around a circle track with an angular velocity of 10 degrees per second. It would take the car how long to drive one "lap"? About 3.6 seconds right?

Remember that there are 360 degrees in a circle. So, \[Angle ~ displaced = \omega \cdot t ~\left [\rm degrees \over time \cdot time \right] \]where \(\omega\) is the angular velocity expressed as\[\omega = { \rm degrees~ or~ radians \over time }\]

The same approach applies here.

We need to find when the two cars meet on the circle.

Remember that a circle has 2 pi radians. If the cars meet, they would have travelled a combined distance of 2 pi radians.

This is expressed as

Total angular displacement = angular displacement of car 1 + angular displacement of car 2.

\[\theta_T = \theta_1 + \theta_2\]\[2 \pi = \theta_1 + \theta_2\]

Recall that\[\theta = \omega \cdot t\]\[2 \pi = \omega_1 t + \omega_2 t\]

Remember, that when the cars meet, they will have been travelling for the same amount of time. \[2 \pi = (\omega_1 + \omega_2) \cdot t\]

Answer 6

Oops! It takes the car in my example 36 seconds to drive around the circle. Not 3.6 seconds.

Answer 7

Oh, their combined distance travelled would only be pi though since they started on opposite sides of the track.