# 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.

## Answers and Replies

Stephen Tashi
Science Advisor
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
Science Advisor
Homework Helper
Gold Member
2020 Award
I can make no sense of your calculation. How is g involved? Why do you divide by 32? Is a acceleration and v velocity? If so, how can a=v?
Some tips:
- work entirely symbolically, not plugging in numbers until the final step
- define your variables
- symbolic variables have dimension, but not units; constants have units. Thus, m=64000 is wrong, m=64000 lb can be right. a=v is wrong, a= v / (1 hour) can be right.