# Find its velocity as a function of time?

## Homework Statement

A boat weighs 64,000 lb. Its propeller produces a constant thrust of 50,000 lb and the water exerts a resistive force with magnitude proportional to the speed, with k=2000 lb-s/ft. Assuming that the boat starts from rest, find its velocity as a function of time, and find its terminal velocity.

F=mg
F=ma

## The Attempt at a Solution

F=mg
F=ma
m=64000/32=2000
a=2000v/2000=v
dv/dt=-g-v
dv/dt=-32-v
-dv/(32+v)=dt
-ln abs(32+v)=t+C
ln abs(32+v)=-t+C
32+v=Ce^-t
v=Ce^(-t)-32
v(0)=0
C=32
v=32(e^(-t)-1)
The answer in the book is v=25(1-e^(-t)); 25 ft/s. So where did I got wrong? Please correct me.

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Stephen Tashi
In this problem, the force of gravity is balanced by the upward force of water on the boat. So gravity has nothing to do with the motion, which is only in the horizontal direction. I don't see that any of your work incorporates the statement that "the water exerts a resistive force with magnitude that is proportional to speed [of the boat]". Start your work with an equation that gives the net force on the boat in the horizontal direction.

haruspex