Question

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?

Answer 1

t = 1.068 s

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)

`(dv)/(dt)= - kv`

`(dv)/(v)= - kdt`

integrating both side

`\int_(800)^(700)(dv)/(v)= - k\int dt`

`ln((7)/(8))=-kt`

`t = -(1)/(k)ln((7)/(8))`..........(1)

since,

`v = v_1e^(-kt)`

`(ds)/(dt) = 800e^(-kt)`

`\int ds= 800\int_0^t e^(-kt)dt`

`1400= -(800)/(k)[e^(-kt)-e^0]`

from equation 1

`k= -(8)/(14)[(7)/(8)-1]`

`k= (1)/(8)`

putting value of k in equation (1)

`t = -8ln((7)/(8))`

t = 1.068 s