Integrating to determine speed as a function of time

In summary, the conversation discusses the use of Newton's second law and the equation for resistive force, F = -bv, to solve a problem involving an object's initial velocity and the resistive force acting on it. It is determined that the equation in question is ∫ (1/v) dv, which can be integrated to ln(v).
  • #1
AryRezvani
67
0

Homework Statement



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Homework Equations



The above formulas

The Attempt at a Solution



I'm lost on where to start with this. The object has an intial velocity in the X direction and has the resistive force of the plontons acting upon it when it lands. What exactly is the equation located in the problem?
 
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  • #2
Hi AryRezvani! :smile:
AryRezvani said:
What exactly is the equation located in the problem?

That's good ol' Newton's second law, F = ma

F = -bv, ma = mdv/dt, so mdv/dt = bv, so dv/v = -b/m dt :wink:

(the "m =" appears to be a misprint)
 
  • #3
tiny-tim said:
Hi AryRezvani! :smile:That's good ol' Newton's second law, F = ma

F = -bv, ma = mdv/dt, so mdv/dt = bv, so dv/v = -b/m dt :wink:

(the "m =" appears to be a misprint)

Thanks for the response Tiny-Tim :)

Okay, so i follow you somewhat. F = -bv (general formula for resistive force).

According to Newton's second law, F=ma which can be rewritten as F=m(dv/dt).

You then equate those two, and you get m(dv/dt)=-bv.

What happens after this? (dv/v) is the derivative of velocity with respect to velocity? :eek:
 
  • #4
AryRezvani said:
(dv/v) is the derivative of velocity with respect to velocity? :eek:

ah, no …

∫ dv/v is a short way of writing ∫ (1/v) dv …

just integrate it! :smile:
 
  • #5
Ohh so when you integrate that you get ln(v)?
 
  • #6
AryRezvani said:
Ohh so when you integrate that you get ln(v)?

yes! :smile:

(to be precise, ln(v) - ln(vo))
 

FAQ: Integrating to determine speed as a function of time

What is integration in the context of determining speed as a function of time?

Integration is a mathematical process that allows us to find the area under a curve. In the context of determining speed as a function of time, integration helps us find the total distance traveled by an object over a given time period.

Why is it important to integrate to determine speed as a function of time?

Integrating allows us to find the exact distance traveled by an object, rather than just an estimate. This is important in many scientific fields, such as physics and engineering, where precise measurements are crucial.

What is the difference between integrating to determine speed and differentiating to determine acceleration?

Integrating is the reverse process of differentiation. While integrating helps us find the total distance traveled, differentiation helps us find the rate of change of that distance, which is acceleration.

Can integration be used to determine speed for any type of motion?

Yes, integration can be used to determine speed for any type of motion, as long as the speed is changing over time. This includes linear motion, circular motion, and even more complex motions such as projectile motion.

Are there any limitations to using integration to determine speed as a function of time?

Integration relies on the assumption that the speed is changing at a constant rate. This may not always be the case in real-world scenarios, and in such cases, other methods may need to be used to determine speed as a function of time.

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