Question

Suppose that the acceleration of a model rocket is proportional to the difference between 160 ft/sec and the rocket's velocity. If it is initially at rest and its initial acceleration is 280 ft/sec22, how long will it take to accelerate to 128 ft/s

Answer 1

Answer:

Explanation:

Given ,

dv / dt = k ( 160 - v )

dv / ( 160 - v ) = kdt

ln ( 160 - v ) = kt + c , where c is a constant

when t = 0 , v = 0

Putting the values , we have

c = ln 160

ln ( 160 - v ) = kt + ln 160

ln ( 160 - v / 160 ) = kt

(160 - v ) / 160 = $e^{kt}$

1 - v / 160 = $e^{kt }$

v / 160 = 1 - $e^{kt }$

v = 160 ( 1 - $e^{kt }$ )

differentiating ,

dv / dt = - 160k $e^{kt }$

acceleration a   = - 160k $e^{kt }$

given when t = 0 , a = 280

280 = - 160 k

k = - 175

a = - 160 x - 175 $e^{kt}$

a = 28000 $e^{kt}$

when a = 128  t = ?

128 = 28000 $e^{kt}$

$e^{kt }$ = .00457