# Solving Boat's Motion Under Retarding Force

• cscott
In summary, the boat in this problem experiences a retarding force that causes it to slow down after reaching a certain speed. Solving the differential equation using Maple gives a formula for the boat's velocity as a function of time, but this formula can lead to confusion if not interpreted correctly. It is important to pay attention to the values of the variables in order to avoid taking the logarithm of a negative number.
cscott

## Homework Statement

boat turns off engine when it's attained $v_0$ at $t = 0$. Starting from that moment the boat is slowed by retarding force $F = Ce^{-kv}$

## The Attempt at a Solution

$$m\dot{v} = -Ce^{-kv}$$

Throwing this in Maple gives me

$$v(t) = \frac{1}{k}\ln\left(\frac{-kC(t + C^{'})}{m}\right)$$

This doesn't make sense to me--you can't take the log of a negative number and even if I remove that negative from the initial DE v(t)->inf as t->inf

What makes you think this involves the logarithm of a negative number? Whether -kC(t+ C') is negative or not depends on C'. In particular, you know that when t= 0, v= v0. Putting this into your formula
$$v(0)= v_0= \frac{1}{k}ln\left(\frac{-kC(C')}{m}\right)$$
which gives
$$\frac{-kC(C')}{m}= e^{kv_0}$$
or
$$C'= -\frac{m}{C}e^{kv_0}$$
which is negative. You will be taking the logarithm of a positive number as long as t< -C'. But when t= -C', what happens?

That's the problem with using Maple, or a computer or any such rather than doing it yourself! If you have done the rather simple integral yourself you wouldn't have had that question.

EDIT: thinking

Last edited:
You missed the 'k' when solving for C'

$$C' = -\frac{m}{kC}e^{kv_0}$$

Think I got it

Last edited:
Thanks

## What is "Solving Boat's Motion Under Retarding Force"?

"Solving Boat's Motion Under Retarding Force" is a scientific concept that involves using mathematical equations and principles to determine the motion of a boat moving against a resistive force, such as friction or air resistance.

## What factors affect a boat's motion under retarding force?

The factors that affect a boat's motion under retarding force include the mass of the boat, the speed at which it is moving, and the strength of the retarding force itself. Other factors such as the shape and design of the boat can also play a role.

## How is retarding force calculated in boat motion?

Retarding force in boat motion can be calculated using the formula F = μN, where F is the retarding force, μ is the coefficient of friction, and N is the normal force. The coefficient of friction is determined by the materials in contact and the normal force is the perpendicular force applied to the surface of the boat.

## What is the difference between retarding force and propelling force?

Retarding force is a force that acts against the motion of a boat, while propelling force is a force that moves the boat forward. Retarding forces include friction and air resistance, while propelling forces can be generated by engines, oars, or wind.

## How can the motion of a boat under retarding force be predicted?

The motion of a boat under retarding force can be predicted by using Newton's laws of motion and applying them to the specific situation. This involves calculating the net force on the boat and using equations to determine its acceleration, velocity, and position over time.

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