Velocity – A Child's view of the World

Hi guys,

As I mentioned certain scientific terms in my previous post, i would like to go in depth on those concepts,beginning with terminal velocity, it being the most fundamental concept in my post.

So what is terminal velocity?

Terminal velocity is the velocity of an object when the drag force (dependent on the fluid the object is travelling through) acting upon it is equal to the downward force of gravity acting upon it. Simply put, when the air resistance of a falling object cancels out the gravitational force which is pulling it downwards and accelerating it.

So how do these forces affect the motion of the object? The forces cancelling each other out make the object remain at a constant rate of motion.

You may ask why does the object still move when the forces cancel each other out. This is due to the fact that in the beginning the force of gravity still manages to overcome the drag force, allowing the object to gain speed (accelerate) initially. But as the object increases in velocity, the drag force increases, this effect can also be seen in the case of friction (Drag and friction are pretty much the same thing). Lets assume that a boy is dragging a heavy box, full of files, across a distance of 100 meters, now we will imagine this scenario in two different ways, firstly in the case whereby the boy is walking slowly and in the second whereby the boy is running. So in the first case the boy walks, when he reaches the end, he feels the bottom of the box, where the box and the floor meet, it still feels the same as before, now in the second case, he runs, he once again feels the bottom of the box, this time it feels warmer than before. So what can we infer from this scenario? Before I reveal the answer, i would like to state a few properties of friction:

Friction opposes motion
Friction causes wear and tear
Friction produces heat when kinetic energy is converted into thermal energy

So what can we infer? In the second scenario, there was more heat, therefore we can assume that there was more frictional force produced in the second case.

Now lets go back to what i mentioned previously, air resistance increases (Drag Force) as the object’s velocity increases. As seen in the example above, we can tell that this statement is true.

That’s essentially the definition of terminal velocity. Before we move on, lets do a recap:

Terminal velocity is the velocity an object is at when the gravitational force acting upon it is equal to the drag force acting upon it in the opposite direction therefore cancelling out all forces therefore having a resultant force of 0
The drag force acting upon the object increases as the object accelerates due to the downward force of gravity.

Ok so lets move on to the math behind terminal velocity and some examples of it.

The formula for terminal velocity is as follows: $V_t= \sqrt{\frac{2mg}{\rho A C_d }}$

Vt=Terminal Velocity

m=Mass of falling object

g= Acceleration of the object due to gravity

ρ= Density of fluid which the object is travelling through

A= Projected area of the object

Cd= Drag Coefficient

I went through everything in my previous post but lets do a recap on these terms:

mass= Amount of matter in an object (SI Unit : kg)

Acceleration = Rate of increase of speed (SI Unit: m/s^2)

Gravitational force= how much gravitational force does an object exert on another object (SI Unit: N/kg)

Density= Mass per unit volume of matter (SI Unit: kg/m^3)

Projected Area= Area of a falling object which is in contact of the air flowing through it. (SI Unit: m^2)- Same as Area

Drag coefficient= A value which depends on the shape of an object and can only be calculated by using the drag force of the object and other factors or by doing actual testing. This value has no units.

That’s all the terms. So now I shall be doing some examples.

So assuming I drop a metal cube which has a mass of 3 kg and has a projected area of 1 m^2 on Earth 90 degrees downward, through air at a temperature of 25 degrees Celsius, what would the Terminal velocity of the cube be?

All we have to do is input all the values into the formula. The acceleration due to gravity on earth is 9.81 m/s^2. The density of air at 25 degrees Celsius is 1.1839 kg/m^3 and the drag coefficient of a cube is 1.05 facing downward. The result is : 6.881101581m/s.

So there’s Terminal Velocity for you!

I would like to thank Mr Tan Ping Hock and Mr Yao Zhi Wei Adrian, My current and previous physics teachers respectively for clearing my doubts about certain concepts within this topic of terminal velocity!

Thanks for reading!

Clyde Lhui

References:

http://en.wikipedia.org/wiki/Drag_coefficient

http://en.wikipedia.org/wiki/Density_of_air

http://en.wikipedia.org/wiki/Terminal_velocity

Hi everyone!

Recently I have been playing this game called League of Legends (LoL). I’m pretty sure that loads of guys out there play LoL. So I was checking out some info about the game. So in this game, there are 3 maps or Fields of Justice. The first one is Summoner’s Rift, the second being Twisted Treeline and the third is called Howling Abyss. But wait, I’m not here to tell you about LoL. If so why would I label it under ‘Science’? Ok here comes the interesting part. In the description of Howling Abyss LoL said that at screams could be heard in the wind and those were rumoured to be those of warriors who got pushed into the bottomless abyss. So I was thinking “Since its a bottomless pit, the warriors who fell in the abyss wont hit the ground and die so how long will it take for them to die and if they screamed when they died, how long would the screams take to reach the top?”

Ok so the first part’s really simple: How long would they take to die? Assuming there are no flying creatures that eat them up, they did not cry, sweat, pee or breathe through their mouth and they were regular humans, they would take 3 days to die of dehydration.

So that’s part 1 answered, now for part 2. Here’s where it starts to get difficult. How long would their final scream take to reach the top? firstly we have to find out how far down they would have travelled. So assuming they were falling like a skydiver (horizontally) we can find the projected area or the area which is in contact of the air below them. The average male Body Surface Area ( BSA )is 1.9 m^2 therefore to calculate the projected area, we can take the BSA divided by 2 as only one side of their bodies are in contact with the air. therefore the projected area would be 0.95 m^2. Next we have to find their mass. Assuming they were very fit and they had the most ideal body weight according to the Body Mass Index (BMI) and they were average American males in the United States of America who have an average height of 1.763 m, they would have a mass of around 150 pounds or 68.0389 kg. Now we have all the data we need to calculate the terminal velocity of the warriors or the speed they will be at when they stop accelerating. The formula for terminal velocity is as follows: $V_t= \sqrt{\frac{2mg}{\rho A C_d }}$

Vt= Terminal velocity

m=Mass of falling object

g= Acceleration due to gravity

p(rho)= Density of fluid which the object is travelling through (in this case, air)

A= Projected area

Cd= Drag coefficient

So what do all these mean? well we have been through practically everything except for the drag coefficient. So what is drag coefficient? Drag coefficient is a value which cannot be calculated. It is the value which puts the force generated by drag into account. The drag coefficient of a man is 1.0-1.3. Lets assume that our warriors have a drag coefficient of 1.2, not too skinny yet not too fat. The acceleration due to gravity on Earth is 9.8m/s^2 and the density of air at room temperature (25 degrees Celsius) is 1.1839 kg/m^3 so with everything in SI units we can calculate the terminal velocity of the warriors. The final answer is 54.44492062 m/s assuming all our calculations are right and in SI units. this can be rounded off to 54.4m/s (3 s.f.). The next part that we have to solve is the acceleration period. so from 0m/s to 54.4 m/s how long would it take? Once again assuming that the acceleration due to gravity is around 9.8 m/s^2 as on earth, all we have to do is divide 9.8m/s^2 from 54.4 m/s. The result we arrive at is 5.551020408 s. to find the distance travelled during this time, all we have to do is imagine that all this data is being plotted on a distance time graph and find the area during the acceleration period.

1/2*5.551020408 s*54.4 m/s =150.9877551 m

This can once again be rounded off to 151 m ( 3 s.f.)

Now lets move on to the part where we deal with the duration of the freefall. Assuming that they took EXACTLY 3 days to die of dehydration, they would take 24 h*3= 72 h to die. Now we minus the 5.551020408 s from the 72 h. 72 h = 72 h*60 min = 4320 min =4320 min* 60 s= 259200 s

259200 s – 5.551020408 s = 259194.449 s (my calculator only has this many decimals)

Therefore we can assume that our warriors remained at terminal velocity for 259194.449 s.

So to find the distance they travelled at terminal velocity, all we have to do is:

259194.449 s* 54.4m/s= 14100178.02 m

So our warriors travelled a total distance of:

14100178.02 m + 151 m = 14100329.02 m

Finally to calculate the time taken for the sound to reach the top, we have to divide the distance travelled by the speed of sound which is Mach 1 or 346.13 m/s at room temperature (25 degrees Celsius).

14100329.02 m / 346.13 m/s = 40737.09018 s

That is how long the sound would take to reach the place where our warriors fell from. Converting it to hours : 11.3159 h

So there’s the answer!

If you have anything else, like a science question of some sort or you found a mistake in my calculations and want me to rectify it, please post it in the comments section or you could send me a message via the contact me form!

Thanks for reading!

Clyde Lhui 🙂

References:

http://en.wikipedia.org/wiki/Drag_coefficient

http://en.wikipedia.org/wiki/Body_mass_index

http://en.wikipedia.org/wiki/Human_height

http://en.wikipedia.org/wiki/Terminal_velocity

http://en.wikipedia.org/wiki/Body_surface_area

http://www.sengpielaudio.com/calculator-speedsound.htm

http://en.wikipedia.org/wiki/Density_of_air

http://en.wikipedia.org/wiki/Gravity_of_Earth

http://en.wikipedia.org/wiki/International_System_of_Units