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A commuter train travels between two downtown stations. Because the stations are only 1.00 km apart, the train never reaches its maximum possible cruising speed. During rush hour the engineer minimizes the travel interval Δt between the two stations by accelerating for a time interval Δt1 at a1 = 0.100 m/s2 and then immediately braking with acceleration a2 = -0.440 m/s2 for a time interval Δt2. Find the minimum time interval of travel Δt and the time interval Δt1.

So far what I have is that

[tex] a_{1} t_{1} = a_{2} t_{2} [/tex]

and

[tex] x_{f} = 1000 = \frac{1}{2} a_{1} t_{1}^2 + a_{1} t_{1} + a_{2} t_{2}^2 [/tex]

I solved for T1 and substituted it in the second equation and got T1=159.59s and T2=36.27... no need to say those are wrong. I guess I am at a loss, I have been working this problem and the next for about 3 hours and I cannot find the correct solution.

Problem 2

Kathy Kool buys a sports car that can accelerate at the rate of 5.40 m/s2. She decides to test the car by racing with another speedster, Stan Speedy. Both start from rest, but experienced Stan leaves the starting line 1.00 s before Kathy. Stan moves with a constant acceleration of 3.70 m/s2, and Kathy maintains an acceleration of 5.40 m/s2.

(a) Find the time it takes Kathy to overtake Stan.

(b) Find the distance she travels before she catches him.

(c) Find the speeds of both cars at the instant she overtakes him.

Now on this problem I thought I understood it... I was wrong.

I took the integral of the acceleration to get a V equation

[tex] \frac{1}{2} 3.7t_{1}^2 = \frac{1}{2} 5.4t_{2}^2 [/tex]

and I assumed [tex] t_{1} = t_{2} - 1 [/tex]

and substituted in again to get T=5.81s ... which was wrong, and unfortunately I have used all my attempts for option A of this question, so to proceed I really need help on it.

Thank you very much in advance if you take the time to help me.