Equation of Motion: Solving for Mass, Velocity, & Force F(v)

In summary, the student is trying to solve an equation for v as a function of x, but is confused about what to differentiate with respect to. They eventually get it to work using the chain rule.
  • #1
kala
21
0

Homework Statement


A mass m has speed v0 at the origin and coasts along the x-axis in a medium with force F(v). Use the chain rule of differentiation to write the equation of motion in the separated form m*v*dv/F(v)=dx.


Homework Equations


F(v)= -c(v^3/2)


The Attempt at a Solution


So far, i drew a diagram to show what was happening. I know that F(v) is the drag force. I'm really just confused about what the equation of motion is. I know that it is going to have a velocity and mass, I think.
Can anyone help?
 
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  • #2
The equation of motion is just Newtons law,
[tex]F(v) = m a[/tex]
where a is the acceleration
[tex]a = \frac{dv}{dt} = \frac{d^2s}{dt^2}[/tex].
 
  • #3
Okay, that is what i have done so far, is this right,
I have m*v0= -c*v3/2.
then i am suppose to differentiate this using the chain rule which my book is calling the v*dv/dx rule to get a separated form m*v*dv/F(x)=dx. But i don't know what to differentiate with respect to, v or x, but my equation doesn't have an x. So maybe my equation is wrong, I am not quite sure.
 
  • #4
How do you get the right hand side? Don't use any specific form for F(v), just the equation I gave you.
Then, in m*a, replace a = dv/dt by something which looks like what you want (try dv/dt = dv/dt * something), using the chain rule.
 
  • #5
So even though the drag force = F(v) i don't sub that part in?
 
  • #6
No, because there is also an F(v) and not c and v^(3/2) in your final answer.
 
  • #7
so in the end when i use the chain rule should I end up with the drag force equation? Sorry I am still a little confused.
 
  • #8
No, they want you to show how you can go from
F(v) = m a
to
F(v) = m v (dv/dx)
using the chain rule, and then rewrite this to
m*v*dv/F(v)=dx.

The point being that the last line is a differential equation for v as a function of x with separated variables (all the v's on one side, all the x's on the other) so you can presumably solve it more easily than solving
m x''(t) = F(x'(t))
for x as a function of t.
 
  • #9
ok, so understand what i was suppose to do, and got it to work, so was there any reason why they told me what the drag force was?
 
  • #10
Probably in the next question they are going to ask you to solve the equation by integrating both sides :smile:
 
  • #11
they do ask me that, so do i substitute that in now for F(v)
 
  • #12
never mind i totally got it to work! thank you for all of the help!
 

1. What is the equation of motion and what does it represent?

The equation of motion is a fundamental equation in physics that describes the relationship between mass, velocity, and force. It represents Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

2. How do you solve for mass in the equation of motion?

To solve for mass in the equation of motion, you would rearrange the equation to isolate the mass variable. This can be done by dividing both sides of the equation by acceleration and then multiplying by velocity squared. The resulting equation would be: m = F(v)/a. This gives you the mass of an object based on its velocity and the force acting on it.

3. How do you solve for velocity in the equation of motion?

To solve for velocity in the equation of motion, you would rearrange the equation to isolate the velocity variable. This can be done by dividing both sides of the equation by mass and then multiplying by acceleration. The resulting equation would be: v = F/m. This gives you the velocity of an object based on the force acting on it and its mass.

4. How do you solve for force in the equation of motion?

To solve for force in the equation of motion, you would rearrange the equation to isolate the force variable. This can be done by multiplying both sides of the equation by mass and then dividing by velocity squared. The resulting equation would be: F = m x a. This gives you the force acting on an object based on its mass and acceleration.

5. What are the units of measurement for mass, velocity, and force in the equation of motion?

The units of measurement for mass, velocity, and force in the equation of motion depend on the system of measurement being used. In the SI system, mass is measured in kilograms (kg), velocity in meters per second (m/s), and force in Newtons (N). In the US customary system, mass is measured in pounds (lb), velocity in feet per second (ft/s), and force in pounds-force (lbf). It is important to use consistent units when using the equation of motion to solve for these variables.

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