Two soccer players, Mary and Jane, begin running from nearly the same point at the same time. Mary runs in an easterly direction at 3.61 m/s, while Jane takes off in a direction 63.3o north of east at 5.44 m/s. How long is it before they are 23.6 m apart?How far apart are they after 4.17 s

To solve this problem, we can use the vector addition method. Let's break down their velocities into their horizontal and vertical components.

Mary's velocity:
- Horizontal component: 3.61 m/s (since she runs in the easterly direction)
- Vertical component: 0 m/s (since she runs horizontally)

Jane's velocity:
- Horizontal component: 5.44 m/s * cos(63.3)° (since she runs at an angle of 63.3° north of east)
- Vertical component: 5.44 m/s * sin(63.3)° (since she runs at an angle of 63.3° north of east)

Now, let's calculate the time it takes for them to be 23.6 m apart.

1. Calculate the horizontal and vertical separations at this given time.
- Horizontal separation: distance = (horizontal component of Jane's velocity) * time
- Vertical separation: distance = (vertical component of Jane's velocity) * time

2. Use the Pythagorean theorem to calculate the total separation using the horizontal and vertical separations:
- Total separation = sqrt((horizontal separation)^2 + (vertical separation)^2)

Now, let's calculate the time it takes for them to be 23.6 m apart.

1. Calculate the horizontal and vertical separations at this given time.
- Horizontal separation: distance = (horizontal component of Jane's velocity) * time
- Vertical separation: distance = (vertical component of Jane's velocity) * time

2. Use the Pythagorean theorem to calculate the total separation using the horizontal and vertical separations:
- Total separation = sqrt((horizontal separation)^2 + (vertical separation)^2)

Given:
- Distance = 23.6 m

1. Calculate the horizontal separation:
- Horizontal separation = (5.44 m/s * cos(63.3°)) * time

2. Calculate the vertical separation:
- Vertical separation = (5.44 m/s * sin(63.3°)) * time

3. Use the Pythagorean theorem to calculate the total separation:
- 23.6 m = sqrt((Horizontal separation)^2 + (Vertical separation)^2)

Square both sides to eliminate the square root sign:
- (23.6 m)^2 = (Horizontal separation)^2 + (Vertical separation)^2

4. Substitute the values of horizontal and vertical separations into the equation:
- (23.6 m)^2 = [(5.44 m/s * cos(63.3°)) * time]^2 + [(5.44 m/s * sin(63.3°)) * time]^2

5. Square and multiply the values inside the brackets:
- 553.696 m^2 = (29.88672 m/s)^2 * time^2 * cos^2(63.3°) + (29.88672 m/s)^2 * time^2 * sin^2(63.3°)

6. Simplify the equation:
- 553.696 m^2 = (29.88672 m/s)^2 * time^2 * (cos^2(63.3°) + sin^2(63.3°))

Since cos^2(63.3°) + sin^2(63.3°) = 1, the equation becomes:
- 553.696 m^2 = (29.88672 m/s)^2 * time^2

Now, we can solve for the time:

1. Divide both sides by [(29.88672 m/s)^2]:
- time^2 = 553.696 m^2 / [(29.88672 m/s)^2]

2. Take the square root of both sides:
- time = sqrt(553.696 m^2 / [(29.88672 m/s)^2])

3. Calculate the time:
- time = sqrt(0.6288 s^2) = 0.7923 s

Therefore, it takes approximately 0.7923 seconds for them to be 23.6 m apart.

To find how far apart they are after 4.17 seconds, we can use the same approach:

1. Calculate the horizontal separation:
- Horizontal separation = (5.44 m/s * cos(63.3°)) * 4.17 s

2. Calculate the vertical separation:
- Vertical separation = (5.44 m/s * sin(63.3°)) * 4.17 s

3. Use the Pythagorean theorem to calculate the total separation:
- Total separation = sqrt((Horizontal separation)^2 + (Vertical separation)^2)

Substitute the values and calculate the total separation after 4.17 seconds.