A trooper is moving due south along the freeway at a speed of 18 m/s. At time 
t = 0,
 a red car passes the trooper. The red car moves with constant velocity of 29 m/s southward. At the instant the trooper's car is passed, the trooper begins to speed up at a constant rate of 1.9 m/s2. What is the maximum distance ahead of the trooper that is reached by the red car?

We can solve this problem by determining when the trooper catches up to the red car. To do that, we'll first write the position functions for both the red car and the trooper as a function of time, then find the time when their positions are equal. Once we have the time when the trooper catches up, we can find the maximum distance by evaluating the difference in their positions before that time.

### Step 1: Write the position functions
1. **Position of the red car:**  
   The red car is moving at a constant velocity of 29 m/s. Its position as a function of time is:
   \[
   x_{\text{red}}(t) = 29t
   \]
   where \( t \) is the time in seconds, and the position is measured from the moment the car passes the trooper.

2. **Position of the trooper:**  
   The trooper starts with an initial velocity of 18 m/s and accelerates at a constant rate of 1.9 m/s². Its position as a function of time can be given by the kinematic equation:
   \[
   x_{\text{trooper}}(t) = 18t + \frac{1}{2}(1.9)t^2
   \]
   where \( t \) is the time in seconds, the initial velocity is 18 m/s, and the acceleration is 1.9 m/s².

### Step 2: Set the positions equal
The red car is ahead until the trooper catches up. To find the time when the trooper catches up to the red car, we set their position functions equal:
\[
x_{\text{red}}(t) = x_{\text{trooper}}(t)
\]
\[
29t = 18t + \frac{1}{2}(1.9)t^2
\]
Simplifying the equation:
\[
29t - 18t = \frac{1}{2}(1.9)t^2
\]
\[
11t = 0.95t^2
\]
Dividing both sides by \( t \) (assuming \( t \neq 0 \)):
\[
11 = 0.95t
\]
Solving for \( t \):
\[
t = \frac{11}{0.95} \approx 11.58 \, \text{seconds}
\]

### Step 3: Find the maximum distance
The maximum distance between the red car and the trooper occurs just before the trooper starts to catch up, which is at \( t = 11.58 \) seconds.

To find this maximum distance, we calculate the positions of the red car and the trooper at that time:
1. Position of the red car:
   \[
   x_{\text{red}}(11.58) = 29 \times 11.58 \approx 335.82 \, \text{meters}
   \]
2. Position of the trooper:
   \[
   x_{\text{trooper}}(11.58) = 18 \times 11.58 + \frac{1}{2}(1.9)(11.58)^2
   \]
   Let's calculate this.

I'll compute this now.The positions of the red car and the trooper at \( t = 11.58 \) seconds are nearly the same:
- Position of the red car: \( 335.82 \) meters
- Position of the trooper: \( 335.83 \) meters

This means the trooper catches up with the red car right at this moment, so the maximum distance ahead of the trooper that the red car reaches is extremely small—approximately \( 0.01 \) meters, which can be considered negligible.

In conclusion, the red car doesn't get significantly ahead of the trooper before the trooper catches up.

Do you need further details, or would you like me to clarify any part of the explanation?

Here are 5 related questions to explore:
1. What would happen if the trooper’s acceleration were higher?
2. How would the problem change if the red car had a variable velocity?
3. What if the initial speed of the trooper were slower than 18 m/s?
4. Can you calculate the time it takes for the trooper to reach a certain velocity?
5. How does air resistance affect the motion of both vehicles?

**Tip:** When solving motion problems, always start by defining the position and velocity functions clearly based on the problem setup.

A trooper is moving due south along the freeway at a speed of 18 m/s. At time t = 0, a red car passes the trooper. The red car moves with constant velocity of 29 m/s southward. At the instant the trooper's car is passed, the trooper begins to speed up at a constant rate of 1.9 m/s². What is the maximum distance ahead of the trooper that is reached by the red car?

Explore how to calculate the maximum distance a red car moves ahead of a trooper accelerating at 1.9 m/s². Using kinematic equations, we solve this problem with detailed steps, including quadratic equation solving and velocity functions. Suitable for high school students in Grades 10-12.

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