Respuesta :
Answer:
[tex]T = 10.43 s[/tex]
Explanation:
During deceleration we know that the deceleration is proportional to the velocity
so we have
[tex]a = - kv[/tex]
here we know that
[tex]\frac{dv}{dt} = - kv[/tex]
so we have
[tex]\frac{dv}{v} = -k dt[/tex]
now integrate both sides
[tex]\int \frac{dv}{v} = -\int kdt[/tex]
[tex]ln(\frac{v}{v_o}) = - kt[/tex]
[tex]ln(\frac{40}{70}) = - k(t)[/tex]
[tex]kt = 0.56[/tex]
Also we know that
[tex]a = \frac{vdv}{ds}[/tex]
[tex]-kv = \frac{vdv}{ds}[/tex]
[tex]\int dv = -\int kds[/tex]
[tex](v - v_o) = -ks[/tex]
[tex](40 - 70)mph = - k (480 ft)[/tex]
[tex]-30 mph = -k(0.091 miles)[/tex]
[tex]k = 329.67[/tex]
so from above equation
[tex]t = \frac{0.56}{329.67} = 1.7 \times 10^{-3} h[/tex]
[tex]t = 6.11 s[/tex]
initially it starts from rest and uniformly accelerate to maximum speed of 70 mph and covers a distance of 220 ft
so we have
d = 220 ft = 67 m = 0.042 miles[/tex]
now we know that
[tex]d = \frac{v_f + v_i}{2} t[/tex]
[tex]0.042 = \frac{70 + 0}{2} t[/tex]
[tex]t = 4.32 s[/tex]
so total time of motion is given as
[tex]T = 4.32 + 6.11 = 10.43 s[/tex]
