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?

d1 = d2

25t = 0.5a*t^2
25 = 0.5a*t
0.002t = 25
t = 12,500 s. = 3.5 h.

To find out how long it takes for Jill to catch up to Jack, we can use the equation of motion. Let's go step by step.

First, we need to determine Jack's head start, which is the distance he travels before Jill starts accelerating. Jack's speed is 25 m/s, so the distance he travels is determined by multiplying his speed by the time it takes for Jill to start catching up.

Let's calculate the time it takes for Jill to start catching up to Jack. Using the equation of motion:

v = u + at

Where:
v = final velocity (Jill's velocity)
u = initial velocity (Jill's initial velocity)
a = acceleration (Jill's acceleration)
t = time

In this case, Jill's initial velocity is 0 m/s (as she starts from rest), and her acceleration is 4.0 × 10^(-3) m/s^2. We need to find the time (t).

Rearranging the equation:

v = u + at
t = (v - u) / a

Substituting the given values:

t = (25 m/s - 0 m/s) / (4.0 × 10^(-3) m/s^2)

Simplifying:

t = 25 m/s / (4.0 × 10^(-3) m/s^2)

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

t = 6250 s^2 / 4.0 × 10^(-3) s^(-2)

t = 6250 / (4.0 × 10^(-3)) s

t = 1,562,500 s

So, it takes Jill approximately 1,562,500 seconds to start catching up to Jack.

Now, we can calculate the distance Jack travels during this time:

Distance = Speed * Time
Distance = 25 m/s * 1,562,500 s

Distance ≈ 39,062,500 meters

Therefore, Jack has a head start of approximately 39,062,500 meters.

Next, we need to find the time it takes for Jill to catch up to Jack, starting from her initial velocity of 0 m/s and accelerating at 4.0 × 10^(-3) m/s^2. This can be calculated using the equation of motion:

Distance = Initial velocity * Time + 0.5 * Acceleration * Time^2

Where:
Distance = Jack's head start distance (39,062,500 meters)
Initial velocity = Jill's initial velocity (0 m/s)
Acceleration = Jill's acceleration (4.0 × 10^(-3) m/s^2)
Time = Time taken for Jill to catch up

Rearranging the equation:

0.5 * Acceleration * Time^2 + Initial velocity * Time - Distance = 0

Plugging in the values:

0.5 * (4.0 × 10^(-3)) * Time^2 + 0 * Time - 39,062,500 = 0

Simplifying:

2 * (4.0 × 10^(-3)) * Time^2 - 39,062,500 = 0

8 × 10^(-3) * Time^2 = 39,062,500

Time^2 = 39,062,500 / (8 × 10^(-3))

Time^2 = 4,882,812,500 / 8 × 10^(-3)

Time^2 = 610,351,562,500 / 8

Time^2 = 76,293,945,312.5

Taking the square root:

Time ≈ 276,419.3567 seconds

Therefore, it takes Jill approximately 276,419.3567 seconds to catch up to Jack.

Please note that the approximations used in the calculations might introduce slight errors, but this should give you a good estimate of the time it takes for Jill to catch up to Jack.