# Argument with professor: how can you describe the trajectory of a particle?

1. Nov 1, 2009

### kodos

I had a physics test last week. When I got the results, there were some correction mistakes in my opinion, so now it's being revised. I had a small disagreement with my professor over one of the theoric exercises, and tomorrow i'll probably be having the same argument again, so I thought of asking for some help with the concept of trajectory.

(I don't have the test, but this is what I remember of the exercise)

The main exercise was a simple problem in two dimensions in which a person threw a projectile from a certain height with initial velocity in the x axis, with some values given such as the distance travelled, time of flight of the particle, etc. One of the theoric points asked to find an equation that describes the trajectory of the projectile.

What I tried to do was, having the equations that described movement in both axis, x(t) and y(t), somehow come up with an expression for y as a function of x, that would be y(x), but I couldn't do it. So instead I wrote this:

x(t)

y(t)

with the correspondent equations for both, and then I wrote

r(t)= (x(t), y(t))

And I marked this last function as the answer.

Before I saw the correction I asked one of the assistants, who said:

"y(x) does not describe the trajectory. This describes the trajectory (and he threw an object through the air)"

That had not make sense to me, and when I saw the correction, i understood he was obviously wrong.

The correction said that I did not find y(x), and it was marked wrong.

The professor said to me:

"If you'd marked y(t) and x(t) as the answers, it would've been OK too. But you marked the position as the answer"

In my opinion, trajectory can be described in many ways, and the vector function r(t) that gives the position of the particle is one of them.

Does r(t) describe the trajectory?

Also, can r(t) be considered an equation?

2. Nov 1, 2009

### blkqi

Yes, r(t) is a parametric equation, and if you made your calculations correctly it will trace out the trajectory of the projectile, its position over time.

3. Nov 1, 2009

### Staff: Mentor

I agree, r(t) is a parametric equation describing the trajectory.

In the future, if you wish to find y(x) given y(t) and x(t) the approach is simply to solve x(t) for t and then substitute that into the expression for y.

4. Nov 1, 2009

### kodos

Thanks, I'll talk to my professor tomorrow and post how it goes.

5. Nov 1, 2009

### Bob_for_short

r(t) is not a trajectory because it contains a certain t. It is the particle position at t.

Trajectory is a whole curve on (X,Y) plane. It is y(x) or x(y) curve. If for a certain x you find the corresponding y on the trajectory, you cannot tell at what time the particle was at this point nor what velocity was there.

Last edited: Nov 1, 2009
6. Nov 1, 2009

### jambaugh

If I were grading your test and hadn't specified any condition on the form of the trajectory then I would give you full credit.

The trajectory is the set of (spatial) points the object occupies over the time of its flight. That set can be expressed in various ways. Yours is a parametric form implicitly stating:
{(x,y): x=x(t), y=y(t) for t in some real interval}

The equation y = f(x) is another (technically parametric form using x as the parameter):
{(x,y): y = f(x), x in some real interval}

(Technically I should go further to indicate the points p(x,y) and not just the coordinates indicating those points).

One could also give x as a function of y (and union the pieces of trajectory together on which it is a function) or use some other parameter besides time.

The critical issue is did you specify the points of the trajectory sufficiently and with the level of exactness expected in the class (e.g. some professors may require you distinguish the points from the coordinate sets). If the teacher is looking for a specific y=f(x) form he needs to make that explicit in the question.

You should also feel free within reason to ask during the test if there are any expectations as to format of answers. At worst the Professor will say "shut up and finish your test" but that's no worse than not having asked. Typically they will either point to (or hint at) overlooked instructions or make a clarification if needed.

Ultimately a professor seeks to assess your knowledge of the material and the test is his means. Always politely argue that you were expressing that knowledge in your answer on specific disputes and not trying to get away with a "technically right" but "for the wrong reasons" debate (which this case does not appear to be).

As an example a question asking for the trajectory of a rooster's egg where the professor meant "hen". "Rooster's can't lay eggs" is technically right but not demonstrating the knowledge of physics requested. At worst the student should answer "Rooster's can't lay eggs but had it been a hen the answer is ..."

7. Nov 1, 2009

### arildno

Hi, kodos!

How does your textbook define "trajectory"?

Remember that in a given introductory course, grades do NOT only reflect whether you've gotten the "right numbers", but also your ability to follow the set of definitions given to you, EVEN IF THOSE DEFINITIONS ARE OF RESTRICTED USAGE.
Lots of students have problems messing up wit their definitions, and te student who shows his ability to apply whatever definitions he's given correctly, deserves credit for that.

Thus, the only way you can get a solid argument against your professor is to show to him that within the textbook, YOUR definition/understanding of the word "trajectory" is defensible.

What your assistants has said doesn't matter in tis context (that's a matter between te professor and the assistant).

8. Nov 1, 2009

### kodos

In my opinion, the question was a generic theoric question, hinting without saying that I should find an y(x). But the thing is that answers in which some people marked x(t) and y(t) as the answer were marked as correct. If this were not the case, I would probably feel less optimistic about this.

Of course, I will ask this as politely as I can, and if I feel that there's still confusion about this, I'll try to ask for a second opinion. That is all I can do.

My textbook does not give a definition, but it says (rough translation from spanish)

"The general equation for the trajectory y(x) can be obtained from the equations y(t), x(t) subtituting the time...etc etc"

It doesn't imply that there are other ways to describe it. My other textbook does not define it either, but it mostly refers to plotting y(t) vs x(t).

But I will make emphasis in that giving the position function could be another way of describing trajectory, and that if giving the parametric equations only is correct, then giving the function which depends on those equations and describes the same curve in space (the same set of points the particle is in) should get the same credits.

9. Nov 1, 2009

### willem2

The question said "find an equation". Your professor is proably of the opinion that the parametric equations are 2 equations.

10. Nov 1, 2009

### rcgldr

Seems very nit picky, and obviously not related to the "correctness" of your answer.

I consider this to be a single equation for a point in 2d space versus time using an ordered pair. a, b, c, and d are constants.

p(t) = (a t, b t2 + c t + d)

11. Nov 1, 2009

### mikelepore

Just in case people here want a check on the way the word "trajectory" is used in some samples of literature, Serway, _Fundamentals of Physics_, section 3.3, section "Projectile Motion", says "x(t)=.......; y(t)=.......; the time can be eliminated to give the trajectory equation: y(x)=........."

12. Nov 2, 2009

### mikeph

I am unsure.. given that t is a general point in the domain, r fully represents the trajectory, no one is saying that t is a single point.

13. Nov 2, 2009

### Bob_for_short

That is why there are different notions of position and trajectory. It is true that the trajectory (line) can be drawn with varying t. But having a line drawn on a paper in (X,Y) coordinates prevents you from knowing the "local time" and "local velocity".

14. Nov 2, 2009

### rcgldr

Do a web search for "ballistics trajectory", probably the most common real world usage of "trajectory", and you usually find parametric equations. Using order pairs or triplets is an alternative way of combining these into a "single" equation. I thought you were in a physics class, not a semantics debate class.

15. Nov 2, 2009

### Bob S

If the professor gave you specific numbers, then you should write down the general equation for a parabola, find the specific values for the three constants, find the initial vertical velocity, find the time to reach the maximum height, then calculate the time of flight, and the horizontal distance traveled.
Bob S

16. Nov 7, 2009

### kodos

In case someone wants to know, I spoke to my teacher. I got partial credit for the exercise. She made emphasis in the fact that the position r(t) was like a single vector, and that the set of all position vectors gives you the trajectory, which can be obtained from y(x). I told her that I thought that r(t) wasn't just one vector, but a function that describes the set of vectors and it describes the curve y(x), so it should describe the trajectory.

But she also pointed to the fact that in physics, trajectory usually is given by y(x) and that was the point of the exercise. People who gave the equations x(t) and y(t) were given partial credits too, so I guess it's OK but I still disagree in the way she explained what was the position function.