# projectiles, independence of horizontal and vertical motion

#### rodders2357

Hi All

I know there is a nice experiment where you project a 'bullet' horizontally and drop one at the same time, the two land at the same time. This demonstrates the independence of horizontal and vertical motion and allows us to separate the motion when working on projectile problems.

However, i do not understand WHY this works and WHY i am allowed to do this?
Is it because they are right angles to each other so simply cannot affect each other?

#### topsquark

Forum Staff
Is it because they are right angles to each other so simply cannot affect each other?
Exactly. You can take this as a basic principle or you can use linear algebra to get a deeper understanding. However just because linear algebra is a nice way to show this property, scientifically speaking we always have to fall back on the principle "it works so we're going to use it."

-Dan

#### rodders2357

Thank you. Out of interest how would you use linear algebra?

#### Woody

You can think of the bullet trajectory as two separate problems.

Imagine a person standing directly in line with the direction of firing
all they can see is the up and down motion of the bullet,
they cannot see how far forward it has gone
(Ok they can see the apparent size of the bullet change and deduce the distance from that, but lets not get too pedantic).

Imagine a second person looking down from space,
They can see how far forward the bullet has gone, but not how far it has dropped.

Each of these people can separately make a mathematical model of what they can see.
The up and down observer sees the bullet dropping from a rest position under gravity.
The space observer sees the high speed bullet slowing down due to air resistance.

You need both these equations to properly define the trajectory.

Note that the mathematical model created by the up and down observer
is exactly the same as if the bullet had just been dropped.

#### rodders2357

That's an interesting way to look at it!
Thank you.

#### topsquark

Forum Staff
Thank you. Out of interest how would you use linear algebra?
You would set up a basis (sort of like coordinate axes) and analyze the motion in terms of the basis. It's a more precise way of looking at the problem. (It's also helpful in understanding how to change from one coordinate system to another.)

-Dan

#### rodders2357

Ok. So is it about trying to show two vectors are linearly independent and they clearly are like i & j ? ( non-zero determinant?)

#### studiot

Ok. So is it about trying to show two vectors are linearly independent and they clearly are like i & j ? ( non-zero determinant?)

Linear algebra is overkill for projectile problems.

Remember that there is no requirement for basis vectors to be orthogonal.
They just can't be parallel.
If they are EDIT orthogonal /EDIT <parallel> the set of unit vectors is called an orthonormal basis.
This is usually the most useful to work with, but not always.

However we know by definition that horizontal and vertical are orthogonal directions.

So any vector pointing vertically will be orthogonal to any vector pointing horizontally.

Now possible vectors pointing vertically (horizontally)

Velocity
Momentum
Acceleration

And that which is often forgotten

The unit (vertical) vector.

Now if we multiply unit vector by a scalar coefficient we get a vector pointing
In the direction of the unit vector.

In this way we can make distance into a vector.

Of course we can also regard velocity etc as coefficient multiples of the one unit vector.

Does this help?

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1 person

#### rodders2357

Perhaps. I'll muse over it. Thanks

#### studiot

Please note my edit I just noticed a howler in my post#8.
.

Sorry