How Long Does It Take a Person to Reach Terminal Velocity?

[Big-Daddy]

I want to estimate how long it will take a person (I could specify their dimensions and density :P but maybe just take it as 170 cm tall, 80 kg, etc.) to fall a certain height in the gravitational field of the Earth, not neglecting air resistance.

I'm looking at heights anywhere from say 30 m to 10,000 m although I appreciate at both extremes the model I'm expecting wouldn't be that accurate (presumably g may begin to differ with height once you get to 10,000m changes?).

 

[Matterwave]

The terminal velocity will depend very much on how this person falls, how much of his body is exposed to the air resistance. Think of someone falling foot down making a straight vertical line, this person would experience much less air resistance than someone falling face flat towards the ground, sprawled out. The terminal velocity will depend on this orientation (as well as other factors like the clothes this person is wearing, etc.) So, it's hard to give you any concrete answers. But I have heard that roughly the terminal velocity of a human in a flat position is ~100mph, but don't quote me on this.

 

[Big-Daddy]

The terminal velocity will depend very much on how this person falls, how much of his body is exposed to the air resistance. Think of someone falling foot down making a straight vertical line, this person would experience much less air resistance than someone falling face flat towards the ground, sprawled out. The terminal velocity will depend on this orientation (as well as other factors like the clothes this person is wearing, etc.) So, it's hard to give you any concrete answers. But I have heard that roughly the terminal velocity of a human in a flat position is ~100mph, but don't quote me on this.

I did some digging around and came up with an estimate of around 120 mph for a human "in random positions". How about we assume this for terminal velocity?

Clearly once the human is "at" or "very near" terminal velocity, we can model speed/distance/time very easily by taking the speed as nearly constant. But it's not nearly so obvious how the approach to terminal velocity can be modeled in terms of time and distance (or for that matter speed)?

 

[Chestermiller]

You need to formulate Newton's second law for the falling object, and you need a relationship for the drag force acting on the object as a function of the velocity. The constants in the relationship for the drag force must be such that, if you set the acceleration equal to zero, you predict a velocity of 100-120 mph. You can then solve the resulting differential equation for the velocity as a function of time, starting with zero velocity downward at time zero.

Chet

 

[PJC01]

I would approach this question by first understanding the concept of terminal velocity. Terminal velocity is the maximum speed that an object can reach when falling through a fluid, such as air. It occurs when the force of gravity pulling the object down is equal to the force of air resistance pushing against it.

To estimate the time it would take for a person to fall a certain height, we would need to consider the equations of motion with air resistance. This would involve taking into account the person's dimensions and density, as well as the air density and viscosity. However, for the sake of simplicity, we can use the average values for a human of 170 cm tall and 80 kg in our calculations.

Using these values, we can estimate the terminal velocity of the person using the formula Vt = mg/cd, where m is the mass of the person, g is the acceleration due to gravity (9.8 m/s^2), and cd is the drag coefficient of the person. The drag coefficient is a measure of how streamlined an object is and can vary depending on the shape and orientation of the object.

Once we have calculated the terminal velocity, we can then use the equation d = 1/2 * g * t^2 to solve for the time it would take for the person to fall a certain distance, where d is the distance and t is the time.

At a height of 30 m, the time it would take for the person to fall would be approximately 2.2 seconds. At a height of 10,000 m, the time would be approximately 50 seconds. However, as mentioned, this is just an estimate and may not be entirely accurate due to factors such as changing air density and the individual's position and orientation during the fall.

It is also worth noting that at higher altitudes, the value of g may be slightly different due to the Earth's shape and rotation. However, for the given heights, this difference would be negligible and would not significantly affect the estimated time of fall.

In conclusion, by considering the concept of terminal velocity and using equations of motion with air resistance, we can estimate the time it would take for a person to fall a certain height.