Answer:
The position is x=38.33[m]
Explanation:
We take the initial position x = 10 [m] when t = 0.
We must be clear about the concept that if we have a graph that shows the velocity with respect the time, the area under the curves of that graph will show us the displacement of the particle.
Therefore we will determine the areas of each of the geometrical figures are shown in the graph.
Let's remember that the area of a rectangle is:
[tex]A_{rec} = b*h\\where \\b=base\\h=height[/tex]
And the area of a triangle is given by the mass by the height divided between two
[tex]A_{triangle} =\frac{a*b}{2} \\a=heigth\\b= base[/tex]
In this graph the bases of each of the geometrical figures are given by time and height is given by speed.
[tex]x_{2} -x_{0} = \frac{5*2}{2} ; x_{0}=10[m] ; x_{2} =15[m]\\[/tex]
[tex]x_{6} -x_{2} = ((6-2)*5)+\frac{((8-5)*(6-2))}{2} ; x_{6} =41[m][/tex]
Now the area under the curve after six seconds is negative, as the speeds are negative as shown in the graph.
In the graph we can see that the velocity is not an integer value when the time is 8 [s], but we can determine this velocity using the properties of similar triangles.
[tex]\frac{4}{3} =\frac{v_{8} }{2} \\\\v_{8} = \frac{4*2}{3} \\v_{8} = 2.66[m/s]
[tex]x_{8} -x_{6} =-\frac{2.66*2}{2} ;x_{8}=38.33[m][/tex]
