Respuesta :
Answer:
The distance the train travels between 0 and 0.2 hours is 17.78 miles
The distance the train travels between 0.2 and 0.4 hours is 6.8376 miles
Step-by-step explanation:
v₁ = 160 mi/hr at t = 0
Deceleration a = 1280 ( 1+ 4t)^(-3)
The velocity at a future time is given by
[tex]v(t) = v(0)+\int\limits^t_0 a(t) \, dt[/tex]
Therefore, plugging the values we get
[tex]v(t) =160+\int\limits^t_0 -1280(1+4t)^{-3} \, dt[/tex]
Which gives
[tex]v(t) =160(1+4t)^{-2}[/tex]
Similarly, the position of the vehicle is given by
[tex]s(t) = s(0)+\int\limits^t_0 v(t) \, dt[/tex]
Which gives s(0) = 0 and [tex]v(t) =160(1+4t)^{-2}[/tex]
Therefore,
[tex]s(t) = 0+\int\limits^t_0 160(1+4t)^{-2}\, dt[/tex]
Which gives
[tex]s(t) = \frac{160t}{4t+1}[/tex]
Therefore, the distance traveled between points 0 and 0.2 is given by
[tex]s(0.2)-s(0) = \frac{160\times 0.2}{4\times 0.2+1} - 0 = 17.78 \, miles[/tex]
Similarly the distance the train travels between 0.2 and 0.4 seconds is
[tex]s(0.4)-s(0.2) = \frac{160\times 0.4}{4\times 0.4+1}-\frac{160\times 0.2}{4\times 0.2+1} = 6.8376 \, miles[/tex].
