Question

Describe a situation in which neglecting the effects of air resistance would lead to significantly wrong predictions.

Short answer

A napkin in a free fall is a situation in which neglecting the effects of air resistance would lead to significantly wrong predictions.

Step-by-step solution

Definition of Air resistance

Air resistance is a type of friction that happens when air collides with another object (a force that resists motion). It's the force that an item feels as it travels through the air. The two permanent forces of nature that move on any item on Earth are air resistance and gravity.

Expression for air resistance force

In order to determine in which circumstances it is not reasonable to neglect the effects of air resistance, the definition of the air resistance force must be considered.

${\mathrm{F}}_{\mathrm{air}}=\frac{1}{2}{\mathrm{C\rho Av}}^2$

From this expression, it can be concluded that the effects of air resistance cannot be neglected in the following situations:

• Objects with large area (high value of A ).

• Objects moving at a very high speed (high value of V ).

Objects with a high drag coefficient (high value of C ). This depends on the shape of the object and its interaction with the surrounding air.

Deducing the situation for neglecting air resistance

A less obvious way of increasing the effects of air resistance is by diminishing the mass of the object.

Write the equation for the scenario for an object in free fall.

$$\begin{gathered} \underset{}{\sum {\mathrm{F}}_{\mathrm{y}}=\mathrm{mg}-\frac{1}{2}{\mathrm{CpAv}}^2} \\ =\mathrm{ma} \end{gathered}$$

Simplify right sides of the equal to sign.

$\mathrm{g}-\frac{{\mathrm{C\rho Av}}^2}{2\mathrm{m}}=\mathrm{a}$

When $\mathrm{m}<<1\mathrm{kg}$then$\mathrm{g}-\frac{{\mathrm{C\rho Av}}^2}{2\mathrm{m}}\approx -\frac{{\mathrm{C\rho Av}}^2}{2\mathrm{m}}$ this implies that$\mathrm{a}\approx -\frac{{\mathrm{C\rho Av}}^2}{2\mathrm{m}}$ .

Therefore, another possible situation in which it is not reasonable to neglect air resistance are as follows:

• In free fall: very light objects.

• Any combination of the situations mentioned.

The obtained analysis could become quite complex if it is considered that the objects have characteristics that support neglecting air resistance, and others that oppose neglecting it.

However, there are other circumstances in which all the characteristics oppose (or support) neglecting air resistance. A napkin in free fall is a good example. It's mass is very low and has a relatively large area, therefore it wouldn't be reasonable to neglect air resistance in this occasion.

Therefore, the situation is a napkin in free fall.