NuclearVision
Hello guys,
So we were taught by our physics teacher an odd method to find the equation of the trajectory. I used the books method, it seemed quiet simple, and less complicated. Briefly, it can be told as follow:
According to newtons 2nd law: sum(forces)=m.a where forces are vectors and a is the acceleration vector.
Considering that the projectile is thrown from x,y=0,0 in an orthogonal plane (o,i,j) and with initial velocity V0, we can write:
Sum(forces)=m.a
m.g=m.a (g= gravitational acceleration, the force pulling weights down to the center of the earth)
g=a
a=-gj then we can determine velocity by integrating the acceleration and the trajectory equation by integrating the velocity.
This will lead us to values totally independent of the weight of the projectile.
I'm sure you all aware that throwing an empty box from a high building with same initial velocity and same environmental conditions won't lead to same results as throwing the same box when it's not empty.
How do you explain this contradiction?
Is it due to some Air Force ?
Share your opinion.
ballad
It is preferable if you try it with a ball because the geometry of the box is unpredictable and the ball will have equal distribution of air friction.
Try it with a marble and a basketball and they will both land at the same time
NuclearVision
Yeah but why does it happen, how is it physically explained, why the trajectory the acceleration the velocity are independent of the weight.
ballad
the empty box will start spinning since the air friction is higher than its weight's ability to compensate the friction whereas the heavy box will will fall straight down.
NuclearVision
Thanks ballad.
But How is it mathematically expressed, the equation of an empty box, a basketball, and the equation of marble ball/heavy box, (trajectory equations) are the same as the only parameters are V0 modulus and its direction and I assume both can be achieved for both heavy and empty box.
Edit: the object is not in a straight free fall.
Let's discuss this on a basis assuming that trajectory land-land parabolic.
ballad
you can't express the empty box falling with an equation because it is just pure randomness.
It may fall in 5secs or in 10secs from the same height, the weird design of the box may act like a parachute.
the mathematical expression might be the 2nd law of Newton: mg-Ftraction=ma
NuclearVision
Okay got that part, what if we are dealing with 2 non-null masses M and 5M what's the difference in their movement.
I hope you're not bored or bothered by my curiosity xD, and thanks again : )!
ballad
not at all, we just having fun
if you are throwing the 2 masses in a parabolic trajectory, apply 2nd law of newton
mg = ma (vector ofc)
g=a=9.81m/s so the mass is independent of the trajectory, and any body you throw will behave the same
NuclearVision
Yeah but does it really happen? Let's think of it: same initial velocity same acceleration different mass leads to same trajectory? Kind of unacceptable.
And if so what compensates the difference in mass to lead to same trajectory?
Ra8
You forgot also the
terminal velocity which is the maximum speed a certain mass can travel in free fall in a certain fluid. This Wikipedia page and set of equations pretty much explains all those concepts you are asking about. But try to get a physics book to understand more...
ballad
it's totally acceptable and there is no conspiracy behind it
It is important to note that the Range and the Maximum height of the Projectile does not depend upon mass of the trajected body. Hence Range and Max. Height are equal for all those bodies which are thrown by same velocity and direction. Air resistance does not affect displacement of projectile.
Wikipedia http://en.wikipedia.org/wiki/Projectile_motion
NuclearVision
Thanks guys I appreciate your help.
I wonder why neither the teacher or the book mention those things.
a-l
actually there is something wrong with the method used :
as you said, F and a are vectors by putting mg=ma you have taken the Y components of those vectors and hence you have calculated the Y component of the acceleration and of course by integrating you'll get the Y component of V; so what you need to do is draw a free body diagram, project on both axis ( take into account the angle that will play a major roll in the maximum height), you'll get a bigger expression of V which will depend on g/ theta/and time(the g dependence come from mg=ma) you differentiate this expression with respect to time and you'll get the expression for a.
http://outreach.phas.ubc.ca/phys420/p420_00/darren/web/noangle/velposnoangle.html
this shows the steps.
PS: there is some YouTube videos that will answer your questions in a very good and simplified way.
NuclearVision
@ A.L g is always vertical and so is a. In a=-gj a and j are vectors. Plus g is oriented to the center of the earth so vect(g)=-||g||vect(j)
Projecting to the x axis is irrelevant, both projections of both vectors(a and g) are null. However I did project v0 to both axis so I could integrate a and get the velocity (v=integral(a)+v0) v0x=(v0.cos a)i, V0y=(v0.sin a).j (-> all are vectors v,v0,v0x,v0y,i,j,intergal(a))
My question is why( the Range and the Maximum height of the Projectile does not depend upon mass of the trajected body)?
Which was included in the wiki article ballad included.
Thanks all.
Metalloy
It does depend.
in the textbook they neglect the effect of air from the equation which is not the real case.
Weight ( mass) is only irrelevent in Vacuum, otherwise we can neglect the force of air.
the force of air depends on the shape of the projectile ( aerodynamics) and mass.
if the projectile is a ball, it can have low air resistance compared to a feather or paper.
but to focus now on the effect of mass, here is how the correct equation will be:
sumF = ma
mg + F(air) = ma [ vectorial equation]
now in the last equation, you can no longer cancel the mass factor amd hence the greater the mass the greater the acceleration.
hope this helps.
Metalloy
It does depend.
in the textbook they neglect the effect of air from the equation which is not the real case.
Weight ( mass) is only irrelevent in Vacuum, otherwise we can neglect the force of air.
the force of air depends on the shape of the projectile ( aerodynamics) and mass.
if the projectile is a ball, it can have low air resistance compared to a feather or paper.
but to focus now on the effect of mass, here is how the correct equation will be:
sumF = ma
mg + F(air) = ma [ vectorial equation]
now in the last equation, you can no longer cancel the mass factor amd hence the greater the mass the greater the acceleration.
hope this helps.
Metalloy
sorry for similar posts, i thought the first one wasnt successfully posted due to the browser crashing.
NuclearVision
Thanks mettaloy now it makes sense, I can imagine a 2T and 2kg object with the same trajectory in vacuum.
But not in the air.