Solution

On the time interval from 0 s and up to 6 s, a cart moves equidistantly with constant acceleration $a=100$ m/s$^{2}$. The initial velocity is 0. The functional dependence of velocity on time on this interval will be:

$$\upsilon(t)=\upsilon(0)+at=100t$$

This is the equation of an increasing straight line.

During the interval from 6 s to 13 s, the cart moves at a constant speed, which it acquired at 6 seconds, which will be the maximum speed of the cart:

$$\upsilon(6)=\upsilon_{max}=100\cdot 6=600 \text{ м/с}$$

On the time interval from 13 s and up to 16 s the cart moves equidistantly with constant acceleration $a=-200$ m/s$^{2}$. The initial velocity is equal to 600 m/s. The functional dependence of velocity on time in this interval will be:

$$\upsilon(t)=\upsilon_{нач}+at=600-200t$$

This is the equation of a decreasing straight line. I note that here time $t$ varies in the range from 0 to 3.

The path traveled by the cart in the interval from 0 s to 6 s has the following functional dependence:

$$S(t)=\upsilon(0)t+\frac{at^{2}}{2}=\frac{100t^{2}}{2}=50t^{2}$$

It is a parabola with a branch pointing upward. At this acceleration interval, the cart has traveled the distance:

$$S(6)=50\cdot 36=1800 \text{ м}$$

On the section with constant velocity 600 m/s (from 6 s to 13 s), the functional dependence of the path on time is an increasing straight line:

$$S(t)=600\cdot t$$

I also note that here $t$ varies in the range from 0 to 7. The path traveled on this interval: $S(7)=600\cdot 7=4200 \text{ m}$

The path traveled by the cart on the interval from 13 s to 16 s has the following functional dependence:

$$S(t)=600t-\frac{200t^{2}}{2}=600t-100t^{2}$$

It is a parabola with a branch pointing downward. The time varies between 0 and 3. In this braking section, the cart has traveled a distance:

$$S(3)=600\cdot 3-100\cdot 9=900 \text{ м}$$

Total path: $S=1800+4200+900=6900 \text{ m}$. Braking of the cart occurred in the section from 6000 m to 6900 m.

The area under the acceleration graph has a mathematical meaning of velocity, so in order to be sure that the cart has stopped it is necessary to check the equality of areas on the acceleration graph above and below the $t$ axis.