Why a heavier skier/whatever is faster, the answer.

[re1s]

This doesn't directly belong in homework help, but I was trying to answer this question and realized it's rarely well answered.

The question is, why does a heavier ball, heavier skier, cyclist, fall faster? Many people seem to believe they will fall at the same speed, as acceleration due to gravity is the same for all of them, however common sense suggests this is wrong. A metal feather WILL fall faster than a real one. Why is this?

The simple answer is air resistance, however how we get to this takes a bit longer. We may also ask the question, doesn't friction also play a part?

With the example of a skier:

The acceleration of the skier at a given time is given by:

$$a = \frac{\sum F}{m}$$

$$\sum F = F_wsin \phi + F_d + F_f$$

Where phi is the angle of the slope, Fd is the drag force (air resistance), and Ff is the frictional force.

$$F_w = mg$$

$$F_d = -\rho A C_d v^2$$

http://en.wikipedia.org/wiki/Drag_equation"

$$F_f = -\mu F_wcos\phi$$

From this we can get:

$$a = \frac{\sum F}{m} = \frac{F_w + F_f + F_d}{m} = \frac{mg\sin \phi - F_w \mu \cos \phi - \rho AC_dv^2}{m}$$

So

$$a = g \sin \phi - \mu g \cos \phi - \frac{\rho A C_dv^2}{m}$$

From this we can see that acceleration due to gravity and frictional force are independent of the mass, but the effect of air resistance on acceleration decreases with a larger mass!

Therefore, a heavier skier will be faster!


This might not be the right forum for this, feel free to move it.
If anyone thinks I'm wrong, or can explain it better, please post!

 

[gneill]

$$a = g \sin \phi - \mu g \cos \phi - \frac{\rho A C_dv^2}{m}$$

but $$\rho = m / V$$
where $$V$$ is the volume of the skier.

So,

$$a = g \sin \phi - \mu g \cos \phi - \frac{ A C_dv^2}{V}$$

which is independent of the mass of the skier.

 

[re1s]

rho is the density of the fluid, in this case the air, not the skier.

It's a constant.

 

[gneill]

rho is the density of the fluid, in this case the air, not the skier.

It's a constant.

Yeah, I realized that about 1 minute before you posted. Mea culpa.

 

[rwisz]

I can confirm that your explanation is correct. The key factor here is air resistance, which is often overlooked when discussing the speed of objects falling or moving through the air. Air resistance is a force that acts in the opposite direction of motion and increases with the square of the velocity. This means that as an object moves faster, the air resistance it experiences increases exponentially.

In the case of a skier, their weight does not affect the force of gravity pulling them down the slope, but it does affect their overall mass and therefore the amount of air resistance they experience. Heavier objects have a greater mass, meaning they require more force to accelerate them. This means that they will experience less acceleration due to air resistance and will therefore move faster than a lighter object with the same amount of force acting on it.

Additionally, as you mentioned, friction also plays a role in the speed of a skier. However, in this scenario, the frictional force is also affected by the mass of the skier. As the mass increases, so does the force required to overcome friction, meaning that a heavier skier will have a greater force acting on them and will therefore be able to overcome friction more easily and move faster.

In conclusion, the weight of an object does have an impact on its speed, contrary to what some may believe. Air resistance and friction are both affected by the mass of an object, and a heavier object will experience less resistance and be able to overcome friction more easily, resulting in a faster speed.