To solve this problem we use the general kinetic equations.
We need to know the time it takes for the car to reach 130 meters.
In this way we have to:
[tex]x(t) = x_0 + v_0t + 0.5at ^ 2[/tex]
Where
[tex]x_0[/tex] = initial position
[tex]v_0[/tex] = initial velocity
[tex]a[/tex] = acceleration
[tex]t[/tex] = time
[tex]x(t)[/tex] = position as a function of time
[tex]130 = 0 + 12(t) + 0.5(2.3)t ^ 2[/tex]
[tex]1.15t ^ 2 + 12t - 130[/tex].
We use the quadratic formula to solve the equation.
[tex]t = \frac{-12 \± \sqrt {(12) ^ 2-4(1.15)(- 130)}}{2 (1.15)}[/tex]
t = 6.63 s and t = -17.1 s
We take the positive solution. This means that the car takes 6.63 s to reach 130 meters.
Then we use the following equation to find the final velocity:
[tex]v_f = v_0 + at[/tex]
Where:
[tex]v_f[/tex] = final speed
[tex]v_f = 12 +2.23(6.63)[/tex]
The final speed of the car is 27.25 m/s
