# Friction problem from Halliday and Resnick

• Farnak
In summary, the conversation discusses a problem involving a boat's deceleration due to kinetic friction and how to solve it using differential equations. The formula for the force of kinetic friction is given and it is explained how this affects the boat's acceleration and velocity over time. The conversation also mentions using integration to solve the equation and the need to know differential equations for a better understanding.
Farnak
Hi, new member here ^_^

"A 1000kg boat is traveling at 90km/h when its engine is shut off. The magnitude of the frictional force f_k between boat and water is proportional to the speed v of the boat: f_k = 70v, where v is in meters per second and f_k is in Newtons. Find the time required for the boat to slow to 45 km/h."

From what I'm getting, the only force on the boat is the force of kinetic friction and as time passes the velocity decreases, so the force of kinetic friction's magnitude will decrease as well. So this will decrease the magnitude of the acceleration and cause velocity to decrease more slowly as time passes by ...

I'm having trouble putting all that together not being used to dealing with non-constant acceleration problems so could someone please help me organize my thoughts for this problem? Thanks!

This can be summarized neatly in the differential equation,

$$F = -kv = ma = m\frac{dv}{dt}$$

where k is the constant of proportionality in the problem. Along with the initial condition v(0) = 90km/h, this equation can be solved to give you v(t). Would you know how to do that? Does this make sense to you?

Are you supposed to solve the equation through integration? If so, I think I'm messing up:

-kv = m (dv/dt)
Integrating both sides
(-k/2)v^2 + C = mv

But an equation like this gives me a constant value for velocity =( ... where did I mess up?

Farnak said:
-kv = m (dv/dt)
Integrating both sides
(-k/2)v^2 + C = mv

-kv=mdv/dt
therefore b*dv/v=dt where b= -m/k
therefore b*loge(vf/vi)=t
where vf and vi are the final n initial velocities...

Ahhh ok, so I guess I need to learn differential equations if I want to read more, sorry for the calculus stupidity, thanks!

## 1. What is friction and how does it affect motion?

Friction is a force that resists the motion of an object when it comes into contact with another object or surface. It acts in the opposite direction of the motion and can cause objects to slow down or stop.

## 2. How is the coefficient of friction determined?

The coefficient of friction is determined experimentally by measuring the force required to move an object across a surface at a constant speed. The ratio of this force to the normal force (the force perpendicular to the surface) is the coefficient of friction.

## 3. What are the different types of friction?

The three types of friction are static friction, kinetic friction, and rolling friction. Static friction occurs when an object is at rest and is trying to be moved. Kinetic friction occurs when an object is already in motion. Rolling friction occurs when an object is rolling over a surface.

## 4. How does friction impact everyday life?

Friction plays a vital role in our everyday lives. It allows us to walk without slipping, write with a pen, and even drive a car. Without friction, we would not be able to perform many tasks that we take for granted.

## 5. Can friction be reduced or eliminated?

Friction can be reduced, but it cannot be eliminated entirely. Lubricants, such as oil or grease, can be used to reduce friction between surfaces. Additionally, smoother surfaces can also reduce friction. In space, where there is no atmosphere, friction is significantly reduced, but it cannot be fully eliminated.

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