Math Is Fun Forum

paulb203

Member

Registered: 2023-02-24

Posts: 321

Gauss and freefalling objects

I’m trying to apply Gauss’s formula for adding consecutive integers to the first second of a freefalling object.
The formula is n(n+1)/2
E.g, to add the numbers 1 to 100 we do,
n=100
100(100+1)/2
=100(101)/2
=10100/2
=5050

*

During the first second of freefall an object, I’m told, averages a velocity of 5m/s downwards (is this the average of Om/s and 5m/s?) *edit*; (is this the average of 0m/s and 10m/s, not the average of 0m/s and 5m/s)
I’m assuming that one way to get this average would be to divide that first second into tenths, add the ten corresponding velocities, and divide by 10.
So for the first tenth we would reach 1/10m/s, for the second tenth we would reach 2/10m/s, etc, up to the tenth tenth where we reach 10/10m/s.
Using Gauss’s formula for the addition we get;
n(n+1)/2
n=10
=10(10+1)/2
=10(11)/2
=110/2
=55
Which we divide by 10 as we’re talking about tenths, so we get 5.5
But the average velocity for the first second of freefall is 5m/s, not 5.5m/s.
Where am I going wrong

Last edited by paulb203 (2024-09-24 22:11:31)

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Prioritise. Persevere. No pain, no gain.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Gauss and freefalling objects

From the (newtonian) equations of motion for a object  falling from rest with  an acceleration g (ok to take this as 10m/s/s) the final velocity is given by v = 0.5gt^2

So when t = 0.1   v = 0.05
              t = 0.2   v = 0.2
              t = 0.3   v = 0.45

................................................

              t = 1     v = 5

The v values aren't in arithmetic progression so that formula won't work.

Bob

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You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Oculus8596

Banned

From: Great Lakes,Illinois

Registered: 2024-09-18

Posts: 126

Re: Gauss and freefalling objects

From the (newtonian) equations of motion for a object  falling from rest with  an acceleration g (ok to take this as 10m/s/s) the final velocity is given by v = 0.5gt^2

So when t = 0.1   v = 0.05
              t = 0.2   v = 0.2
              t = 0.3   v = 0.45

................................................

              t = 1     v = 5

The v values aren't in arithmetic progression so that formula won't work.

Bob

Is this a physics problem? Objects falling implies physics.

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The best things in life are not always free.

paulb203

Member

Registered: 2023-02-24

Posts: 321

Re: Gauss and freefalling objects

From the (newtonian) equations of motion for a object  falling from rest with  an acceleration g (ok to take this as 10m/s/s) the final velocity is given by v = 0.5gt^2

So when t = 0.1   v = 0.05
              t = 0.2   v = 0.2
              t = 0.3   v = 0.45

................................................

              t = 1     v = 5

The v values aren't in arithmetic progression so that formula won't work.

Bob

Thanks, Bob.
Are you sure of that equation?
I thought it was v=gt.
E.g, v=10x1
       v=10m/s
I.e, an object in freefall reaches 10m/s after 1 second (with an average of 5m/s for the period 0-1s)

*

Your equation seems to give that average.

Forgive me if I'm wrong, my brain is starting to hurt after an hour on this

(Just got this from the Google AI overview; To find the final velocity of a free falling object, you can use the formula v = gt)

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Prioritise. Persevere. No pain, no gain.