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Y_Kistochka [10]
4 years ago
12

An object falls freely in a straight line and experiences air resistance proportional to its? speed; this means its acceleration

is ?a(t)=??kv(t), where k is a positive constant and v is the? object's velocity. The speed of the object decreases from 800 ft/s to 700ft/s over a distance of 1400 ft. Approximate the time required for this deceleration to occur?
Mathematics
1 answer:
KiRa [710]4 years ago
5 0

Answer:

t = 1.068 s

Step-by-step explanation:

given,

a(t) =- k v(t)

speed of the object decreases from 800 ft/s to 700 ft/s

distance = 1400 ft

time for deceleration = ?

a(t) =- k v(t)

\dfrac{dv}{dt}= - kv

\dfrac{dv}{v}= - kdt

integrating both side

\int_{800}^{700}\dfrac{dv}{v}= - k\int dt

ln(\dfrac{7}{8})=-kt

t = -\dfrac{1}{k}ln(\dfrac{7}{8})..........(1)

since,

v = v_1e^{-kt}

\dfrac{ds}{dt} = 800e^{-kt}

\int ds= 800\int_0^t e^{-kt}dt

1400= -\dfrac{800}{k}[e^{-kt}-e^0]

from equation 1

k= -\dfrac{8}{14}[\dfrac{7}{8}-1]

k= \dfrac{1}{8}

putting value of k in equation (1)

t = -8ln(\dfrac{7}{8})

t = 1.068 s

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