# Falling bodies and terminal velocity

• CatastrophicF
In summary, two objects with different masses, dropped from the same height at the same time, will fall at different rates when air resistance is taken into account. The equation of motion, ma = mg - f(v), where f(v) is the function for air resistance, shows that the mass of the object does not cancel out and therefore affects the rate of falling. This is why objects with different masses will reach their terminal velocity at different times. This concept was first demonstrated by Galileo, who may have also pointed out that this only holds true in a vacuum.
CatastrophicF
Is it correct to assume that two objects, with different masses, dropped from the same height at the same time will fall at the same rate until one of the objects reaches its terminal
velocity? What are the applicable equation(s) used to calculate this?

Welcome to PF!

Hi CatastrophicF! Welcome to PF!
CatastrophicF said:
Is it correct to assume that two objects, with different masses, dropped from the same height at the same time will fall at the same rate until one of the objects reaches its terminal
velocity?

nooo …

the equation of motion is ma = mg - f(v),

where f(v) is the function giving the air resistance for that particular body at speed v

(usually depends on size and shape but not on mass)

tiny-tim said:
Hi CatastrophicF! Welcome to PF!

nooo …

the equation of motion is ma = mg - f(v),

where f(v) is the function giving the air resistance for that particular body at speed v

(usually depends on size and shape but not on mass)

Thank you for the welcome and response

ma = mg. I understand that equation is essentially telling me that in a vacuum, regardless of mass, two bodies will fall/accelerate at the same rate (on Earth 9.8m/s/s). I think I start to become a little confused once those bodies are falling through a fluid, in this case air. I read and watch videos of people telling me things like the acceleration of gravity is the same on all objects (I'm happy with that) and that intertia is the reason why if I held and dropped a basketball and a bowling ball they would hit the ground at the same time (I'm also happy with that). So my quest to self educate myself in the laws of physics begins with me thinking that mass is a inconsequential property of falling bodies buuuuttttt I also know about terminal velocity, in which mass is a variable. Sooooo my qunadary is as follows; if all bodies accelerate at 9.8m/s/s due to gravity and their terminal velocity is calculated by Cd, surface area and m... why would they not fall at the exact same rate up until the moment one of the objects acheived its terminal velocity?

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Hi CatastrophicF!
CatastrophicF said:
… if all bodies accelerate at 9.8m/s/s due to gravity and their terminal velocity is calculated by Cd, surface area and m... why would they not fall at the exact same rate up until the moment one of the objects acheived its terminal velocity?

because Cd, surface area and m are different (and have different ratios) for different bodies

(in particular, denser bodies of the same size will have the same air resistance, but more weight, and so will fall faster)

btw, technically, nothing ever reaches terminal velocity!

CatastrophicF said:
Thank you for the welcome and response

ma = mg. I understand that equation is essentially telling me that in a vacuum, regardless of mass, two bodies will fall/accelerate at the same rate (on Earth 9.8m/s/s). I think I start to become a little confused once those bodies are falling through a fluid, in this case air. I read and watch videos of people telling me things like the acceleration of gravity is the same on all objects (I'm happy with that) and that intertia is the reason why if I held and dropped a basketball and a bowling ball they would hit the ground at the same time (I'm also happy with that). So my quest to self educate myself in the laws of physics begins with me thinking that mass is a inconsequential property of falling bodies buuuuttttt I also know about terminal velocity, in which mass is a variable. Sooooo my qunadary is as follows; if all bodies accelerate at 9.8m/s/s due to gravity and their terminal velocity is calculated by Cd, surface area and m... why would they not fall at the exact same rate up until the moment one of the objects acheived its terminal velocity?

Look at Tiny Tim's equation in response #3. The drag force is independent of the mass m, so when air drag is present, the mass m no longer cancels from the equation.

So technically speaking a heavier object...will...fall faster than a lighter one. For example if I dropped two bowling balls (same height/same time) that were the same size, one with a mass of 10lbs and one with a mass of 20lbs...the 20lbs bowling ball would, mathematically/physically although maybe not perceivably, fall faster and hit the ground first...?

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CatastrophicF said:
So technically speaking a heavier object...will...fall faster than a lighter one. For example if I dropped two bowling balls (same height/same time) that were the same size, one with a mass of 10lbs and one with a mass of 20lbs...the 20lbs bowling ball would, mathematically although maybe not perceivably, fall faster and hit the ground first...?

yup!

tiny-tim said:
yup!

Thank you so much for helping me work that out. It's amazing that such a common assertion
("heavy and light objects fall at the same rate") is fundamentally incorrect. How did that happen?

galileo first demonstrated it …

i don't know whether he pointed out that it only works exactly in a vacuum

## 1. What is a falling body?

A falling body is any object that is in motion due to the force of gravity. This includes objects that are dropped, thrown, or propelled into the air.

## 2. What is terminal velocity?

Terminal velocity is the highest speed that a falling body can reach when the force of gravity is balanced by the force of air resistance. At this point, the object will no longer accelerate and will continue to fall at a constant speed.

## 3. How is terminal velocity affected by the mass of the falling body?

The mass of a falling body does not affect its terminal velocity. This means that objects of different masses will reach the same maximum speed when falling through a fluid, such as air.

## 4. What factors affect the terminal velocity of a falling body?

The terminal velocity of a falling body is affected by the density and viscosity of the fluid it is falling through, as well as the shape and surface area of the body. Objects with a larger surface area or more streamlined shape will have a lower terminal velocity.

## 5. Can terminal velocity be reached in a vacuum?

No, terminal velocity can only be reached when there is air resistance acting on the falling body. In a vacuum, where there is no air, objects will continue to accelerate until they reach the ground or encounter another force.

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