Problem with Friction dependent on velocity, mostly a math problem.

In summary, the student is having trouble with integrating the homework equation a = -0.98|v|. They try integrating from the start, but run into an error. They then go back and try to solve it for v using the initial conditions v0 = 5 m/s and t0 = 0. They get the result v= 5e-0.98t.
  • #1
Simen
4
0
Hello, first time here for me, I hope I posted in the right subforum.
I have a task at hand, just started a new physics course, but sadly I am far from deft with mathematics, and my physics book is in a different language and uses different notation than I am used to. The part I am having trouble with is the following:

Homework Statement




We have a block, with the starting velocity 5m/s.
The friction from the block is given as a = -μ|v|g
Where μ = 0.1 and g = 9.8 m/s^2

Find v(t)


Homework Equations


v(t) = v(0)+at

a = dv/dt = (d/dt)(dx/dt) =(d2x)/dt2


The Attempt at a Solution



As it stands now, I I realize I should integrate a in order to obtain v(t), but I am horrible at math, and do not know where to start, for some reason I keep ending up with an expression without v at all, but that does not help things either. If someone could show me the first steps in this that would be great. I've written a python program to solve it numerically as well, though I can not check if it is working, because I am too inept to get my analytical solution.

Would something like this be a step in the right direction?
a = -0.98|v|
v(0) = 5 m/s
a(0) = 0

dv/dt = -0.98*v

dv/dt2 = -0.98*v*dt

(dv/dt2)1/v = -0.98dt

And then integrate from here? Could someone show me an example? Doesn't need to have any of my numbers, so long as I can learn some of what I need from it.

If anyone could provide any help, that would be greatly appreciated!
 
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  • #2
Welcome to PF!

Hello Simen! Welcome to PF! :wink:
Simen said:
dv/dt = -0.98*v

(you left out a µ)

Now you "separate the variables" …

dv/v = -0.98µdt …

carry on from there :smile:
 
  • #3
Thanks for the welcome:)

I actually didn't forget the μ, since μ = 0.1 and it was to be multiplied with g = 9.8 I just multiplied them before I started integrating.

Anyways, does this look right?
dv/dt = -0.98v

dv/v = -0.98dt

Integrate:

ln(v)-ln(v0)=-0.98(t-t0)

Clean up a little:
v/v0=e-0.98(t-t0)

v = v0e-0.98(t-t0)

Then putting in t0 = 0 and v0 = 5 and getting
v(t) = 5e-0.98t

Does that look about right?
 
  • #4
Simen said:
v = v0e-0.98(t-t0)

Then putting in t0 = 0 and v0 = 5 and getting
v(t) = 5e-0.98t

(nice formatting, btw! :biggrin:)

Excellent! :smile:

(btw, always best to check by mentally putting the solution back into the original differential equation. to see if it comes out right … which it does! :wink:)
 
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  • #5




Hi there, welcome to the forum! It's great that you are seeking help with your physics homework.

First of all, don't worry if you are not comfortable with mathematics. Physics involves a lot of math, but with practice and patience, you can improve your skills.

Now, let's tackle the problem at hand. The friction force in this case is dependent on the velocity of the block, which means it will change as the block moves. This makes the problem a bit more complicated, but we can still solve it using basic calculus.

As you have correctly identified, the first step is to integrate the acceleration equation to obtain the velocity equation. So, let's start with the equation you have written:

dv/dt = -0.98*v

We can rewrite this equation as:

dv/v = -0.98*dt

Now, we can integrate both sides of the equation with respect to time:

∫dv/v = ∫-0.98*dt

ln|v| = -0.98t + C

Where C is the constant of integration.

Now, we can exponentiate both sides of the equation to get rid of the natural log:

v = e^(-0.98t+C)

But, we can rewrite the constant C as another constant, let's call it K:

v = e^(-0.98t+K)

Next, we can use the initial condition given in the problem to solve for K. At t=0, v=5 m/s, so we can substitute these values into the equation:

5 = e^(-0.98*0+K)

5 = e^K

Taking the natural log of both sides, we get:

ln(5) = K

So, our final equation for velocity becomes:

v = e^(-0.98t+ln(5))

Or, we can simplify it to:

v = 5*e^(-0.98t)

This is the velocity equation for the block as it moves with friction. You can use this equation to calculate the velocity at any given time. If you need to find the position of the block, you can integrate the velocity equation again with respect to time.

I hope this helps and gives you a better understanding of how to approach problems like this. Remember to always practice and seek help when needed. Good luck with your physics course!
 

What is friction?

Friction is a force that resists motion between two surfaces in contact with each other.

How is friction dependent on velocity?

Friction is directly proportional to the velocity of an object. As the velocity increases, the friction force also increases.

What is the mathematical equation for friction?

The mathematical equation for friction is F = μN, where F is the friction force, μ is the coefficient of friction, and N is the normal force.

How does friction affect motion?

Friction can slow down or stop the motion of an object or it can also cause an object to move in a different direction.

What factors can affect the coefficient of friction?

The coefficient of friction can be affected by the type of surfaces in contact, the roughness of the surfaces, and the presence of any lubricants or contaminants.

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