To solve this problem, let's break it down step-by-step.

Step-by-Step Approach:

1. Understand the Motion:

   • The car starts at rest, accelerates with a constant acceleration $+a_1$ for the first $\frac{1}{4}$ of the total distance $d$.
   • Then, the car decelerates (negative acceleration $-a_2$) for the remaining $\frac{3}{4}$ of the distance, finally coming to rest at $x_f$.

2. Define the Distances:

   • Distance covered during acceleration: $d_1 = \frac{1}{4}d$
   • Distance covered during deceleration: $d_2 = \frac{3}{4}d$

3. Using Equations of Motion:

   • For the first segment (acceleration $+a_1$):

     • Initial velocity, $v_0 = 0$
     • Final velocity after this segment, $v_1$
     • Use $v_1^2 = v_0^2 + 2a_1d_1$

   • For the second segment (deceleration $-a_2$):

     • Initial velocity, $v_1$ (same as the final velocity from the first segment)
     • Final velocity after deceleration, $v_2 = 0$
     • Use $v_2^2 = v_1^2 - 2a_2d_2$

4. Relate the Velocities:

   • Since the car comes to rest at the end of the second segment, $v_2 = 0$, you can substitute and solve for $v_1$.

5. Calculate Time for Each Segment:

   • For the first segment (acceleration phase): Use $v_1 = v_0 + a_1t_1$ to solve for $t_1$.
   • For the second segment (deceleration phase): Use $v_2 = v_1 - a_2t_2$ to solve for $t_2$.

6. Find Total Time:

   • The total travel time $T$ is the sum of the times for both segments: $T = t_1 + t_2$

Your Next Steps:

• Write down the equations for $v_1$, $t_1$, and $t_2$.
• Solve the equations step-by-step using algebra to find expressions for $t_1$, $t_2$, and ultimately the total travel time $T$.

Would you like me to go over any specific step in more detail?

Related Questions to Consider:

1. How would the problem change if the distances covered during acceleration and deceleration were equal?
2. What if the car had an initial velocity instead of starting from rest?
3. How would varying the accelerations $a_1$ and $a_2$ affect the total time?
4. What would be the effect on travel time if there was a third segment with a different acceleration?
5. How can you graphically represent the car's velocity and acceleration over time?

Tip:

Always remember to keep track of units when solving physics problems to ensure consistency and correctness.