# Trajectory Graphs: Direction of Velocity & Acceleration

• mymabelline
In summary, the trajectory graph only shows the direction of velocity, not the magnitude. However, it can still be used to find instantaneous acceleration by looking at the nullclines on a phase space plot. Time information is needed to determine velocity or acceleration, so without equations or time values, the velocity cannot be determined from the graph alone.
mymabelline
So it is my understanding from my book that the information that we can get from a trajectory graph is the direction of velocity, but not the magnitude of velocity. If the tangent line to a point on a trajectory graph just gives the direction of velocity, why can the trajectory graph still be used to find instantaneous acceleration? Or is the acceleration we find with the trajectory graph just the rate of change of the direction of velocity instead of the rate of change in the magnitude of velocity? My main problem is just understanding how ax, ay, a, v, vx, and vy are related to a trajectory graph. I guess I just can't wrap my head around it because I think that we should only be able to find these values from a position vs. time graph.

Is it just that because you must use the acceleration and velocity in the x direction to find the x position and the acceleration and velocity in the y direction to find the y position that you should be able to find these values from the trajectory graph even though the slope of the trajectory graph only tells the direction of velocity?

Well, it depends on what plot you're looking at. If you're looking at the phase space plot, then you only have the x,y positions to see values for.

You would also plot the nullclines of the system on such a phase plot. Those nullclines (which are lines in a two-dimensional system) tell you exactly where the rate change (dx/dt and dy/dt) are zero. This would be velocity if your system pertains to motion.

So if your trajectory is on either side of the nullcline, this tells you whether it is a positive rate change or a negative rate change (i.e. a positive or negative velocity would indicate direction). And the farther away from the nullcline, the greater the value of that change rate.

But because the nullclines are somehow parameterized on the phase plot (i.e. they're not linear with the variables x and y), you can imagine that the real valued change rates would also have to be parameterized on the line. There's no easy way to do that. You would have to plot all the "non-nullcline" lines. And each of the fixed points are going to have different strengths, which you can only analyze by computing the eigenvalues of the jacobian. So this is not so trivial.

If you want to know the velocity or acceleration, you would need information about time. A graph that has only x and y (and perhaps z) values does not have time information.

so the graph can't show time, but isn't time a parameter in the position equations for the x and y positions? so if i was given at least 2 or 3 of the variables in the position equation for the horizontal and vertical direction i could find the unknown variable using the position equation that described the trajectory graph right?

That would depend on the equation. It may be that a numerical approximation is the only way to solve it.

mymabelline said:
so the graph can't show time, but isn't time a parameter in the position equations for the x and y positions? so if i was given at least 2 or 3 of the variables in the position equation for the horizontal and vertical direction i could find the unknown variable using the position equation that described the trajectory graph right?
In your first post, you said nothing about having any equations, or any time values; only a graph of x & y. If that graph (with no times indicated, and no equations in terms of time) is all you have, you could not determine the velocity.

If you are given time values for points along the graph, or are given the equations explicitly in terms of the time, then you could get the velocity.

## 1. What is a trajectory graph?

A trajectory graph is a visual representation of the path or movement of an object over time. It shows the position of the object at different points in time and can also include information about its velocity and acceleration.

## 2. How is velocity represented on a trajectory graph?

Velocity is typically represented by the slope of the line on a trajectory graph. A steeper slope indicates a higher velocity, while a flatter slope indicates a lower velocity. The direction of the slope also indicates the direction of the velocity, with a positive slope representing movement in the positive direction and a negative slope representing movement in the negative direction.

## 3. What does a flat line on a trajectory graph indicate?

A flat line on a trajectory graph indicates that the object is not moving, as there is no change in its position over time. This could mean that the object is stationary or moving at a constant velocity.

## 4. How is acceleration represented on a trajectory graph?

Acceleration is represented by the change in the slope of the line on a trajectory graph. If the slope becomes steeper, it indicates an increase in acceleration, while a flatter slope indicates a decrease in acceleration. The direction of the slope also indicates the direction of the acceleration, with a positive slope representing an increase in speed and a negative slope representing a decrease in speed.

## 5. What can we learn from a trajectory graph?

A trajectory graph can provide information about an object's position, velocity, and acceleration over time. It can also help in predicting the future movement of the object and identifying any patterns or trends in its movement. Additionally, it can be used to analyze and compare the movement of multiple objects, making it a useful tool in various scientific fields such as physics and engineering.

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