A circular track has several concentric rings where people can run at their leisure. Phil runs on the outermost track with radius rP while Annie runs on an inner track with radius rA = 0.90 rP. The runners start side by side, along a radial line, and run at the same speed in a counterclockwise direction.How many revolutions has Annie made when Annie's and Phil's velocity vectors point in opposite directions for the first time?

Assuming that the runners start at the same point and run at the same speed, Annie will have made one revolution when their velocity vectors point in opposite directions for the first time.

To determine the number of revolutions Annie has made when her velocity vector points in the opposite direction to Phil's for the first time, we need to understand the concept of angular velocity and the relationship between angular velocity and linear velocity.

1. First, let's define some variables:
- rP represents the radius of the outer track (where Phil is running).
- rA represents the radius of the inner track (where Annie is running).
- ωP represents the angular velocity of Phil.
- ωA represents the angular velocity of Annie.

2. The linear velocity of a runner on a circular track can be calculated by multiplying the radius (r) by the angular velocity (ω).
- The linear velocity of Phil can be calculated as: vP = ωP * rP.
- The linear velocity of Annie can be calculated as: vA = ωA * rA.

3. Since Phil and Annie are running at the same speed, their linear velocities must be equal.
- Therefore, vP = vA.

4. We can substitute the formulas for linear velocity from step 2 into the equation from step 3:
- ωP * rP = ωA * rA.

5. We know that rA = 0.90 rP. Substituting this value into the equation gives:
- ωP * rP = ωA * (0.90 rP).

6. Simplifying the equation, we get:
- ωA = (ωP * rP) / (0.90 rP).

7. Since angular velocity represents the rate of change of angle with respect to time, it is measured in radians per second.
- The angle covered in one revolution is 2π radians.

8. The time taken for Annie to run one revolution can be calculated as the angle (2π radians) divided by her angular velocity (ωA):
- Time taken for Annie to run one revolution (TA) = (2π radians) / ωA.

9. Similarly, the time taken for Phil to run one revolution (TP) can be calculated as:
- Time taken for Phil to run one revolution (TP) = (2π radians) / ωP.

10. Since we are interested in the first time their velocity vectors point in opposite directions, we need to find the smallest integer value of TA and TP that satisfies this condition.
- Therefore, we need to find the smallest positive integer value of TA / TP, which represents the ratio of Annie's and Phil's revolution counts.

11. Calculate the ratio of revolution counts:
- Revolution ratio = TA / TP = [ (2π radians) / ωA ] / [ (2π radians) / ωP ].

12. Simplify the equation:
- Revolution ratio = ωP / ωA.

13. Substitute the value of ωA from step 6 into the equation to get:
- Revolution ratio = ωP / [ (ωP * rP) / (0.90 rP) ].

14. Simplify the equation further:
- Revolution ratio = (0.90 rP * ωP) / (ωP * rP).
- Revolution ratio = 0.90.

Therefore, Annie will make 0.90 (or 9/10) of a revolution when her velocity vector points in the opposite direction to Phil's for the first time.

To determine the number of revolutions Annie has made when her velocity vector is opposite to Phil's for the first time, we need to compare their angular velocities.

The angular velocity is defined as the rate of change of the angle with respect to time. It can be calculated using the formula:

ω = v / r

where ω is the angular velocity, v is the linear velocity, and r is the radius of the circular track.

Since both Phil and Annie run at the same speed, their linear velocities will be the same. Let's call this common linear velocity v.

For Annie:
ωA = v / rA

For Phil:
ωP = v / rP

Since both runners start side by side along a radial line, at the beginning, their velocity vectors will be pointing in the same direction.

For Annie's and Phil's velocity vectors to be opposite, the difference in their angles must be 180 degrees.

Let's assume that Annie starts running when Phil has completed x revolutions. The angle, θA, covered by Annie can be calculated as:

θA = ωA * t

where t is the time.

The angle, θP, covered by Phil can be calculated as:

θP = ωP * t

For the first time their velocity vectors are opposite, we have:

θA - θP = 180 degrees

Substituting the expressions for θA and θP:

ωA * t - ωP * t = 180 degrees

Simplifying:

(v / rA) * t - (v / rP) * t = 180 degrees

(vt / rA) - (vt / rP) = 180 degrees

Simplifying further:

vt * (1 / rA - 1 / rP) = 180 degrees

Since Annie's radius, rA, is given as 0.90 times Phil's radius, rP, we can substitute rA as 0.90 rP:

vt * (1 / (0.90 rP) - 1 / rP) = 180 degrees

Simplifying again:

vt * ((1 - 0.90) / (0.90 rP)) = 180 degrees

vt * (0.10 / (0.90 rP)) = 180 degrees

0.10vt / (0.90 rP) = 180 degrees

Multiplying both sides by (0.90 rP):

0.10vt = 180 degrees * (0.90 rP)

0.10vt = 162 degrees * rP

Dividing both sides by 0.10v:

t = (162 degrees * rP) / (0.10v)

Now, we know that the time it takes for both runners to reach the point where their velocity vectors are opposite is given by t. We can calculate the distance covered by Annie, which will be the circumference of the inner track, as:

distance traveled by Annie = 2πrA

Substituting rA = 0.90 rP:

distance traveled by Annie = 2π(0.90 rP)

distance traveled by Annie = 1.8πrP

Since the runners are running at the same speed, we can calculate the total distance covered by Phil as:

distance traveled by Phil = 2πrP

To calculate the number of revolutions Annie has completed, we divide the distance traveled by Annie by the distance traveled by Phil and multiply it by the number of revolutions completed by Phil at that time:

Number of revolutions completed by Annie = (distance traveled by Annie / distance traveled by Phil) * x

Substituting the distances:

Number of revolutions completed by Annie = (1.8πrP / 2πrP) * x

Number of revolutions completed by Annie = (1.8/2) * x

Number of revolutions completed by Annie = 0.9x

Therefore, Annie will have completed 0.9x revolutions when her velocity vector is opposite to Phil's for the first time.