# Solving Boat's Friction Force and Speed

• davidge
In summary: The constant will vary with x. (I confess I am taking a shortcut here, which works for this problem but not in general. In general you need to solve the DE for x(t) and then do the integration. It's just that in this case the DE is separable, so you can solve it without integration.)In summary, the conversation discusses a problem involving a boat with a mass of 1000 kg and a friction force proportional to its speed. The conversation goes on to discuss the integration of the friction force and the use of the equation ##V = V_o + at## to find the time it takes for the boat to slow down from 25 m/s to 12.5 m/s
davidge

## Homework Statement

A boat of mass 1000 kg is moving at 25 m/s. The friction force ##f## is proportional to the speed ##v## of the boat, ##f = 70v##. How many time will take for the boat to reduce its speed to 12.5 m/s?

## Homework Equations

##\vec{F_r} = m \vec{a_r}##

## The Attempt at a Solution

Since ##f## is proportional to ##v## at each instant ##t##, I integrated ##f## to get the total force.
$$f_{\text{total}} = \int_{v_o}^{v_f}-70vdv$$
the minus sign is because that force is opposite to the movement.
Then, I assumed that the total force equals the mass times the total acceleration. Next, I substituted the value for the total acceleration from the above expression and I used it in the equation: ##V = V_o + at## to get the total time ##t##. Is this correct?

davidge said:
f=70v
The 70 has units. The equation ought to be given as f=70v kg/s.
davidge said:
I integrated f to get the total force.
To state your equation in full, ∫f.dv = ∫70v.dv. What is the physical meaning of ∫f.dv? I can't think of one.

Write the differential equation relating velocity to acceleration.

Thanks haruspex.
haruspex said:
What is the physical meaning of ∫f.dv? I can't think of one.
I think it's actually wrong, because it would give units $$\frac{kg}{s} \frac{m²}{s^2}$$ and this is not Newtons.
haruspex said:
The 70 has units. The equation ought to be given as ##f = 70 v \ kg/s##.
Yes, I have forgotten to mention it in the OP post.

haruspex said:
Write the differential equation relating velocity to acceleration.
Would this be
$$v = \int \frac{d^2x}{dt^2} dt$$

davidge said:
Would this be
$$v = \int \frac{d^2x}{dt^2} dt$$
That is a general true statement. I meant the DE (not an integral equation) representing the given problem, using the expression for f.

haruspex said:
That is a general true statement. I meant the DE representing the given problem, using the expression for f.
Ah, ok.

$$f = -70v = ma \\ \Rightarrow a = - \frac{70v}{m}$$

The problem is that ##a## isn't constant, so how can we substitute it in the equation ##V = V_o + at## to solve for ##t##?

davidge said:
a isn't constant
No, but it has a well-known relationship to v. Remember, we are looking for a differential equation.

haruspex said:
No, but it has a well-known relationship to v. Remember, we are looking for a differential equation.
Would this be

$$\frac{d^2x}{dt^2} = - 70 \frac{dx}{dt} \\ \frac{dx}{dt} = -70x \\ \Delta V = -70 \Delta x \\ \Rightarrow \Delta t = \frac{\Delta x}{\Delta v} = - \frac{1}{70}$$

davidge said:
Would this be

$$\frac{d^2x}{dt^2} = - 70 \frac{dx}{dt} \\ \frac{dx}{dt} = -70x \\ \Delta V = -70 \Delta x \\ \Rightarrow \Delta t = \frac{\Delta x}{\Delta v} = - \frac{1}{70}$$
The first integration stage is fine, except that you should allow for a constant of integration. For the second stage you need to rearrange the equation so that dt occurs on one side and only terms involving x (and dx) occur on the other.

haruspex said:
The first integration stage is fine, except that you should allow for a constant of integration. For the second stage you need to rearrange the equation so that dt occurs on one side and only terms involving x (and dx) occur on the other.
Ok. So, $$\frac{d^2x}{dt^2} = -70 \frac{dx}{dt} + a_o \\ \frac{dx}{x} = -70dt \\ lnx = -70t + a_ot + c \\ t = \frac{lnx - c}{(-70 +a_o)}$$ where it would remain to find ##a_o##, ##c## and ##x##... I guess ##a_o## could be taken to be equal to $$- \frac{70v_o}{m}$$

davidge said:
Ok. So,##\frac{d^2x}{dt^2} = -70 \frac{dx}{dt} + a_o ##
No, the constant of integration comes in as you integrate, not before.
davidge said:
## \frac{dx}{x} = -70dt ##
That equation is after integration, and then dividing by x. You need to include the constant as part of the integration step, before dividing by x.

## 1. What is boat's friction force and how does it affect its speed?

Boat's friction force refers to the resistance that the water exerts on the boat as it moves through it. This force directly affects the boat's speed by slowing it down and requiring more energy to maintain a certain speed.

## 2. How can we calculate the boat's friction force?

To calculate the boat's friction force, we need to know the coefficient of friction between the boat and the water, the water density, the boat's speed, and its surface area in contact with water. The formula for calculating friction force is: F = μ * ρ * v^2 * A, where μ is the coefficient of friction, ρ is the water density, v is the boat's speed, and A is the surface area.

## 3. Can we reduce the boat's friction force and increase its speed?

Yes, we can reduce the boat's friction force by using a more streamlined shape, reducing the contact area between the boat and the water, and using materials with lower coefficients of friction. By reducing the friction force, we can increase the boat's speed.

## 4. How does the boat's weight affect its friction force and speed?

The boat's weight does not directly affect its friction force. However, a heavier boat may require more energy to overcome the friction force and maintain a certain speed. Additionally, the weight distribution of the boat can also affect its speed and maneuverability in the water.

## 5. Is there a way to completely eliminate friction force for a boat?

No, it is impossible to completely eliminate friction force for a boat as it moves through water. However, by using advanced materials and engineering techniques, we can greatly reduce the friction force and optimize the boat's speed and performance.

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