Thread: time and distance ..please help with steps.

If A and B are running in a circular track in a direction opposite to that in which C is running, who is running at twice and thrice the speeds of A and B respectively and on the same track. They start running from a same point. It is known that A's average speed is 3 m/s and the track is 120 m.
When after the start, will B find himself equdistant and between A and C for the first time?

Given the speed of A, find the speed of B and C.

The circular track has 2*pi radians and 120 meters long so people running round it change angle at a rate of 2*pi/120= pi/60 radians per meter.
If B is running with speed v then he is changing direction at a rate of v*(pi/60) radians per second.
After a time t B will have gone v*t*(pi/60) radians
Putting the circle in the x-y plane B's x co-ordinate is r*cos(v*t*(pi/60)) where r is the radius of the circle
B's y co-ordinate will be r*sin(v*t*(pi/60))

Likewise if A is running at speed u then his x and y co-ordinates will be r*cos(u*t*(pi/60)) and r*sin(u*t*(pi/60))

C is running in the opposite direction at a speed w. Because its going the opposite direction you have to put a minus sign in front, so C's x co-ordinate is -r*cos(w*t*(pi/60)) and y co-ordinate is -r*sin(w*t*(pi/60))

Now you have their co-ordinates in terms of time get an expression for the distance between their points.
Distance |BC|= |AB|
That gives you one equation in terms of time, solve the equation for t.

Remember all angles are in radians.

Is there a way to solve by just using algebra?

I am sure you would understand the algebraic method and also notice how making a drawing makes things easy to understand.

Hello, hisajesh!

We find that: .

Picture the circular track; circumference 120 meters.
The one-meter marks are labeled clockwise: 1, 2, 3 . . . ending with 120 at the top.
A and B are running clockwise; C is running counter-clockwise, starting at the top.

In seconds, has run to point
In seconds, has run to point
In seconds, has run to point

When is exactly halfway between and ?
When is 's coordinate the average of 's coordinate and 's coordinate?

We have: .

Therefore: .