# Falling Cats

Why does your cat have a better chance of surviving a fall from a large cliff than you do?

- Air resistance determines the terminal velocity that can be achieved for a given cross-sectional area and mass.
- Different configurations of mass and area will have different terminal velocities.

Why does your cat have a better chance of surviving a fall from a large cliff than you? This example incorporates air resistance into free fall to derive the terminal velocity reached. The terminal velocities and physiology of cats and humans are compared.

Suppose you and your cat have the misfortune of falling off of a large cliff. Given that the acceleration is due to the force of gravity acting on both a cat and on yourself during free fall, you might suspect that the velocity at which both of you hit the ground is the same. In this example we will show that this is not the case due to the drag force associated with air resistance. As you fall through the air you experience a drag force

where *A* is your cross-sectional area, *v* is your speed, ρ is the density of air (1.29 kg/m^{3}) and the drag coefficient C_{D} describes how slippery (aerodynamic) an object is. For everyday objects C_{D} is often ≈ 0.51 . Drag (D) is a vector whose direction is opposite to the direction of your motion, with magnitude given above. Notice the drag increases as speed and cross-sectional area increases.

The net force on you as you fall is

Note this is not a situation of constant acceleration, since the acceleration depends on your speed *v*, which changes as you accelerate. However, as you fall, your velocity increases until the upwards drag force on you is the same as the force of gravity. With F_{net} = 0, you stop accelerating and have reached your **terminal velocity**,

For the rest of your fall your velocity will stay constant at this velocity.

What is your terminal velocity? If you weigh 65 kg and you were falling flat, your cross sectional area would be roughly your width times your height

This would give v_{term} = 56 m/s. If you fall feet first that would decrease your area and increase your terminal velocity.

A cat weighs about 9 lbs ≈ 4.1 kg. Its cross sectional area, if it falls flat, is again approximately

This gives v_{term} = 35 m/s. The cat is not travelling as fast as you are when it hits the ground so its deceleration,

is less. This means that the force, *F* = *ma*, felt on impact is also less, assuming the same time to come to a complete stop, Δ*t*.

Because cats have four feet, they can land on their front two feet first and then on their back two feet. This also decreases the force felt upon deceleration by lengthening Δ*t*. Note: this does not imply that cats don't die or get injured from falls of even small heights since they are not always able to land properly on their feet. Even if they were able to land properly, the force that they experience when they land may still break their bones and cause injury.

A more detailed calculation is necessary to determine the distance an object must fall to reach 95% of its terminal velocity. It turns out that a human falling flat would require around 470 m whereas a cat would require 210 m. The major point is that because of drag the velocity of your cat is less than your velocity, even when falling from a distance that is smaller than that needed to reach terminal velocity. For small distances, jumps around 20 m or less, air drag is negligible thus you and your cat would roughly travel at the same velocity. The reason why cats survive those short falls better is their ability to right themselves in the air and land on their four feet.

**Summary:**

When falling in the presence of air there exists a drag force due to air resistance accounting for objects with different shapes and sizes reaching the ground with different velocities. If the distance fallen is large enough, a maximum velocity will be reached called the terminal velocity where the object stops accelerating and will fall at the terminal velocity until impact. A rough calculation showing that cats have a lower terminal velocity partially accounts for why cats have a better chance of surviving a very long fall.

- 1. Knight, R.D, Jones, B. and Field, S.
*College Physics: a strategic approach*(1^{st}ed.). Pearson Addison-Wesley, 2007.

© Physics and Astronomy Outreach Program at the University of British Columbia (Janelle van Dongen 2009-08-17)

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