Effect of air resistance

In summary, air resistance is the force that opposes the motion of an object through air, caused by the collision of air molecules with its surface. It can impact the trajectory of a falling object by slowing it down and changing its course. It should not be ignored when calculating the speed of a moving object because it can significantly affect its speed and acceleration. Air resistance also plays a significant role in the performance of vehicles, particularly in terms of fuel efficiency and speed. However, its effect can be reduced by increasing the object's aerodynamics and using materials with low drag coefficients.
  • #1
affans
11
0

Homework Statement



The effect of air resistance is to slow down an object. It can be shown that the height of a falling object is given by the following:

y=[tex]y_{o}[/tex] - [t + ([tex]e^{-bt}[/tex] - 1) / b] * g/b.

Show that for short times the eqn is reduced to

y=[tex]y_{o}[/tex] - 0.5(g)(t^2)

Homework Equations



I think the regular distance formula yf = yi(t) + 0.5(at^2) is what i need here because the reduced eqn resembles it very cloesely.


The Attempt at a Solution


I've tried to do a lim as t approaces 0 on the first eqn. I've tried to equate the second eqn with the first eqn. I've been at it for a couple of hours now.

ANY help would be appreciated.
 
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  • #2
Try the Taylor expansion of exp(-bt) and ignore terms past t2
 
  • #3
The distance formula yf = yi(t) + 0.5(at^2) is actually the same as y=y0 - 0.5(g)(t^2), the equation the question wants you to prove, since a=-g.

Do you know what the Taylor series expansion of e^x is? If you do, expand e^-bt, discard higher-ordered terms, and y will reduce to y0 - 0.5(g)(t^2).
 
  • #4
Oops, I posted my answer before I saw rock.freak's.
 
  • #5
Hi,
thankyou very much. I have gotten the reduced equation.

but can someone explain to me why the taylor expansion was needed? what does the expansion have to do with "time being very short"?
also, why are only the terms until t^2 needed?

thanks
 
  • #6
affans said:
Hi,
thankyou very much. I have gotten the reduced equation.

but can someone explain to me why the taylor expansion was needed? what does the expansion have to do with "time being very short"?
also, why are only the terms until t^2 needed?

thanks

if t is small, terms like t3,t4 and higher will give even smaller numbers. So depending on the degree of accuracy, these numbers don't affect the desired accuracy.
 
  • #7
so if i have understoon taylor series correctly, it just means the

function e^x (or in my case e^-bt) can be REWRITTEN as a sum of individual terms given by the taylors series. Am i correct?

and if I am correct then in my question, the higher the degree on t, the smaller the number.

so my third question is why stop at t^2?
 
  • #8
"function e^x (or in my case e^-bt) can be REWRITTEN as a sum of individual terms given by the taylors series. Am i correct?"

Yup. The more terms you include, the more accurate the approximation. The Taylor series expansion becomes infinitely accurate with an infinite number of terms.

"and if I am correct then in my question, the higher the degree on t, the smaller the number."

Exactly.

"so my third question is why stop at t^2?"

Because that's the level of approximation that gives you y=y0 - 0.5(g)(t^2). If you include more terms, you'll get a more accurate equation, but it won't be the same as the free-fall equation.
 

1. What is air resistance and how does it affect objects?

Air resistance is the force that opposes the motion of an object through air. It is caused by the collision of air molecules with the surface of the object. The greater the surface area and speed of the object, the greater the air resistance will be.

2. How does air resistance impact the trajectory of a falling object?

Air resistance can cause a falling object to slow down and change its trajectory. This is because the force of air resistance acts in the opposite direction of the object's motion, counteracting the force of gravity.

3. Can air resistance be ignored when calculating the speed of a moving object?

No, air resistance should not be ignored when calculating the speed of a moving object. It can significantly impact the speed and acceleration of an object, especially at high speeds or in highly aerodynamic objects.

4. How does air resistance affect the performance of vehicles?

Air resistance can greatly impact the performance of vehicles, particularly in terms of fuel efficiency and speed. Vehicles with a higher drag coefficient, such as trucks or SUVs, will experience more air resistance and therefore require more energy to maintain speed.

5. Is there a way to reduce the effect of air resistance on an object?

Yes, there are ways to reduce the effect of air resistance on an object. One way is to increase the object's aerodynamics by making it more streamlined. This can be achieved through design and shape modifications. Additionally, using materials with low drag coefficients can also help reduce air resistance.

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