Two girls run around a circular track starting from the same point but in
opposite direction. Girl A and B run at angular speed of 2.2x10-3 rads-1 and 1.4 x 10-3 rads-1 respectively. How long will it take for both of them to meet on the track?
To solve this problem, we can use the concept of relative angular speed. Relative angular speed is the difference in angular speed between two objects moving in opposite directions.
Firstly, we need to determine the relative angular speed of Girl A and Girl B. Girl A's angular speed is 2.2 x 10^(-3) rad/s, and Girl B's angular speed is 1.4 x 10^(-3) rad/s. Since they are moving in opposite directions, we can subtract the two angular speeds:
Relative angular speed = Girl A's angular speed - Girl B's angular speed
= (2.2 x 10^(-3) rad/s) - (1.4 x 10^(-3) rad/s)
= 0.8 x 10^(-3) rad/s
= 8 x 10^(-4) rad/s
Now, we need to determine the time it will take for both girls to meet on the track. To do this, we can use the formula:
Time = 2π / relative angular speed
where π is approximately equal to 3.14159. Substituting the values into the formula:
Time = 2π / (8 x 10^(-4) rad/s)
≈ (2 x 3.14159) / (8 x 10^(-4)) s
≈ 2 x 3.14159 x 10^(4) / 8 s
≈ 2 x 3926.99
≈ 7853.98 seconds, or approximately 2.18 hours
Therefore, it will take approximately 2.18 hours for Girl A and Girl B to meet on the track.
To find out how long it will take for both girls to meet on the track, we need to determine the time it takes for them to complete one full revolution and meet again.
Let's calculate the time taken by Girl A to complete one full revolution (T₁):
T₁ = 2π / ω₁
Where ω₁ is the angular speed of Girl A.
Given, ω₁ = 2.2 × 10⁻³ rad/s,
T₁ = 2π / (2.2 × 10⁻³) = 2π × 10³ / 2.2 ≈ 2857.14 s
Similarly, let's calculate the time taken by Girl B to complete one full revolution (T₂):
T₂ = 2π / ω₂
Where ω₂ is the angular speed of Girl B.
Given, ω₂ = 1.4 × 10⁻³ rad/s,
T₂ = 2π / (1.4 × 10⁻³) = 2π × 10³ / 1.4 ≈ 4494.92 s
Since they are running in opposite directions, they will meet when the time taken by Girl A to complete one full revolution (T₁) and the time taken by Girl B to complete one full revolution (T₂) add up.
Total time taken for them to meet = T₁ + T₂
= 2857.14 s + 4494.92 s
≈ 7352.06 s
Therefore, it will take approximately 7352.06 seconds for both girls to meet on the track.