Jack is driving with a pail of water along a straight pathway at a steady 25 m/s when he passes Jill who is parked in her minivan waiting for him. When Jack is beside Jill, she begins accelerating at the rate of 4.0 × 10−3 m/s2 in the same direction that Jack is driving. How long does it take Jill to catch up to Jack?

To find out how long it takes Jill to catch up to Jack, we need to determine the time it takes for Jill's minivan to cover the initial distance between them and reach Jack's position.

Let's start by finding the initial distance between Jack and Jill when she starts accelerating. Since Jack is already driving at a steady 25 m/s, we can use the equation:

Distance = Speed × Time

The time will be the same for both Jack and Jill when Jill starts accelerating, so we can denote it as 't'. Thus, the initial distance between them is given by:

Initial Distance = Jack's Speed × Time

Initial Distance = 25 m/s × t

Now, let's find out how much distance Jill covers during the time 't' when she is accelerating. We can use the equation of motion:

Distance = Initial Velocity × Time + (1/2) × Acceleration × Time^2

Since Jill starts from rest (zero initial velocity), the equation simplifies to:

Distance = (1/2) × Acceleration × Time^2

Substituting the given values, we have:

Distance = (1/2) × 4.0 × 10^-3 m/s^2 × t^2

Now, let's equate the two distances (Initial Distance = Distance):

25 m/s × t = (1/2) × 4.0 × 10^-3 m/s^2 × t^2

Simplifying the equation, we get:

0.02 × t = 4.0 × 10^-3 × t^2

Dividing both sides by 't', we have:

0.02 = 4.0 × 10^-3 × t

Now, isolate 't' by dividing both sides by 4.0 × 10^-3:

t = 0.02 / (4.0 × 10^-3)

t = 5 seconds

Therefore, it will take Jill 5 seconds to catch up to Jack.

To determine how long it takes Jill to catch up to Jack, we can use the equations of motion.

Let's first calculate the distance Jack will travel before Jill catches up:
The distance Jack travels is equal to his speed multiplied by the time he travels:
Distance traveled by Jack = Speed of Jack × Time = 25 m/s × T

The distance Jill travels can be calculated by the equation of motion:
Distance traveled by Jill = Initial velocity × Time + (1/2) × Acceleration × Time^2
Jill starts from rest (initial velocity is 0), so the equation simplifies to:
Distance traveled by Jill = (1/2) × Acceleration × Time^2

For Jill to catch up, the distance traveled by Jill must be equal to the distance traveled by Jack:
(1/2) × Acceleration × Time^2 = 25 m/s × T

We can rearrange the equation to solve for Time:
(1/2) × Acceleration × Time^2 = 25 m/s × T
(1/2) × 4.0 × 10^-3 m/s^2 × Time^2 = 25 m/s × T
2.0 × 10^-3 m/s^2 × Time^2 = 25 m/s × T
Time^2 / Time = 25 m/s / (2.0 × 10^-3 m/s^2)
Time = (25 m/s) / (2.0 × 10^-3 m/s^2)

Now we can calculate the time it takes Jill to catch up to Jack:
Time = 25 m/s / (2.0 × 10^-3 m/s^2)
Time = 12,500 s / (2.0 × 10^-3)
Time = 6,250,000 s / 1 = 6,250,000 seconds

Therefore, it will take Jill approximately 6,250,000 seconds to catch up to Jack.