Motion in 2 Dimensions

In trying to pin down the location of a particle in a reference frame, we were led into the swamp of vector arithmetic so to speak. Let's try at this point to pick up the thread of our story. We are still dealing with the kinematics of a single particle here but we have relaxed the 1 dimensional restriction applied earlier. For the time being let's restrict ourselves to a two dimensional space since the ideas are the same and the third dimension complicates the pictures. As we did in the 1 dimensional case, we will first work with the general relationships among particle position, velocity and acceleration, then consider the special case of constant acceleration in order to get at some simple equations describing the particle motion.

In our discussion of functions we identified an independent and a dependent variable, then related one to another through a function. This led to a graphical representation of the function. For example we plotted y=10*x-x2. In physics, we are going to want to predict the future so our interest will be in expressing each of the parameters of particle motion as a function of time, making time the independent variable and things like position , velocity and acceleration the dependent variables.

When we introduced the idea of a rate of change of a dependent variable with respect to an independent variable, the initial example was speed being the rate of change of distance with respect to time. So we have already seen one instance of time being the independent variable. Distance and speed are scalar quantities. The vector equivalents are "displacement" and "velocity", including the direction intelligence along with the magnitude. geometric vector subtractionDisplacement is defined as a change in position which we remember is the difference between two vectors, an initial position and a final position. I illustrated this in the vector subtraction display. If the vectors r1 and r2 represent an initial and final position then the vector Dr = r2 - r1 represents a displacement. The use of r to represent position in more than 1 dimension is common in the literature, possibly derived from radius being the distance from a center to a point. Take another look at the Geometric Vector Subtraction display.

How then would we calculate the velocity from the displacement vector? By analogy to the speed formula, we can write a definition of velocity as velocity = displacement / time. Remembering the italics for vectors convention,

v = Dr / Dt .

We know how to divide a vector by a scalar. We just divide each of the vector's scalar components by the scalar t, where the value of t we use is really a Dt, the time difference between when the particle was at r1 and when it was at r2. velocity vector In the next display, Velocity Vector , you may calculate velocities based on a fixed Dt of 2.0 seconds. Use the cursor and mouse button to mark the position vectors r1 and r2.

you should have noticed that the velocity vector was in the same direction as the displacement vector and was half the length. The direction of course was the same as the displacement because division by a scalar does not alter a vector's direction even thought it may reverse the sense of the vector. The length of the velocity vector depends not only on the size of Dt but also on the scale of our reference frame coordinate system.

We have been assuming that the units on our position and displacement vectors were meters so that the values along the axes represented meters of distance. In calculating velocity, we divided by time, yielding units of meters per second. Nothing requires that the same scale on the screen apply to meters per second as to meters but as a matter of convenience unless specified otherwise we will use the same scale factor.

average velocityNow consider a particle moving along some arbitrary path in our reference frame. Since the path is arbitrary we may not assume that the velocity is constant or even changing in any regular way. In the Average Velocity display you will see that the velocity vector calculated over any finite interval of time is really the average velocity over that time. At any instant between the initial and final time, the velocity vector might be different in either direction or magnitude or both. Notice that the length of the displacement vector, Dr, is always shorter than the actual path over which the particle travels. This should tell you something about the relationship of the magnitude of the average velocity and the average speed of the particle.

instantaneous velocityAs you could see by going to smaller and smaller Dt on the Average Velocity display, the velocity approaches a limit as Dt approaches zero. This is the instantaneous velocity of the particle at the point on its path marked by the velocity vector. The direction of the instantaneous velocity is tangent to the curve at that instant. Run the Instantaneous Velocity display to see this illustrated. Notice that the word "speed" is used to denote the magnitude of the velocity.

I indicated that the path of the particle was marked with time ticks because the time dependence of the particle's position was not visible in the chosen reference frame. This is a complication which deserves a bit of exploration.

Suppose that a particle is following some particular path through space. We should make a careful distinction between the graph of y as a function of the independent variable x, and a curve traced out by a moving particle in two dimensional space. The two plots might look the same but in the first case the value of x determines the value of y. In the second case the position of the particle is a function of time. The fact that the position has an x and a y component actually ties the x and y values together so that they are not independent of each other but the relationship is not one of independence and dependence.

What does it mean to say that a position vector, call it r, is a function of time? Well just like y being a function of x meant that there was some rule relating to every x some value of y, r being a function of time means that there is some rule relating to every value of time, t, a value for r. But the vector r may be expressed in terms of its component vectors like this:
r = x * ihat + y * jhat ,

where ihat and jhat are the unit vectors which are constant, meaning they are independent of time. So when we say the position vector r is a function of time we are actually saying that x and y are each functions of time.

In physics we may know, or be able to figure out, what the rule is relating each of the variables x and y to time. We could then plot x as a function of time on one graph and y as a function of time on another using time as the independent variable in each case. Then the relationship between x and y would be fixed, without us ever having to express it directly. This trick of using another variable, time in this case, to get at the relationship between x and y so as to plot the path of a particle is called a "parametric " representation of the particle path and the functions relating x to time and y to time are called parametric equations.

parametric plotTo illustrate the application of parametric equations, let's look at the case of a particle whose horizontal position changes with time like
x = vx * t
and whose vertical position changes with time like
y = vy * t - 4.9 * t2 .

To see the path of such a particle run the Parametric Plot display.

In our parametric plotting example we began with the horizontal and vertical positions as known functions of time. Quite often we will not know these functions explicitly. Later in this course we will be determining the time dependent behavior of various system parameters.

average accelerationNow let's return to looking at our particle on its arbitrary path and explore the acceleration it is experiencing. Remember that acceleration is the rate of change of velocity with respect to time.

a = Dv / Dt

Run the Average Acceleration display to see the relationship among displacement, velocity and acceleration.

The instantaneous acceleration is developed from the average through the limiting process in which we look at smaller and smaller time intervals between adjacent velocity vectors. The Instantaneous Acceleration display shows the instantaneous velocity and corresponding instantaneous acceleration vectors at a series of times along the particle path.

The unit of length is the meter, so displacement is in meters. Velocity is calculated as displacement divided by time so its units are meters per second. Acceleration is calculated as velocity divided by time so its units are meters per second per second, or meters per second squared. Now we will return to the idea of constant acceleration applied to 2 dimensions.

This may be a good place to talk a bit about choosing appropriate reference frames. Careful attention to which way is x can save a lot of work. When we say we will have a constant acceleration in 2 dimensions, that means that the acceleration is constant in both magnitude and direction. We could choose the directions of the axes without regard to the direction of the acceleration vector. If we did, in general the acceleration would have a component along each dimension. Having to keep track of the motion in such an out of whack reference frame adds a lot of complication to the arithmetic.

bad reference frameSuppose for example that the -y axis of the reference frame made an angle q with the direction of the constant acceleration like this. Then the equation describing motion in the x direction would be:

x = x0 + v0x * t + 1/2 * ax * t2 ,

where we have just copied the 1 dimensional constant acceleration results from before. The symbol v0x is the component of initial velocity in the x direction. The symbol ax is the component of the acceleration vector a in the x direction. That component is given by the expression
ax = -|a| * sin(q) .
Putting that all together we get
x = x0 + v0x * t + 1/2 * -|a| * sin(q) * t2 .

Going through a similar process for motion in the y dimension we get:

y = y0 + v0y * t + 1/2 * -|a| * cos(q) * t2 .

good reference frameThe preceeding mess is something we could well do without. Let's see what happens to the these equations when we straighten up the reference frame so that the -y axis lies along the acceleration vector as shown here. The effect of this is to make the angle q equal to zero. Then sin(q) = 0 and cos(q) = 1. Putting those values into the former expressions for motion in the x and y dimensions we get:

x = x0 + v0x * t and y = y0 + v0y * t -1/2 * |a| * t2 .

As you can see, proper choice of reference frame makes life easier. We will return to this idea later.

In our discussion of reference frames I slipped in a significant idea without making much fuss about it. In looking at constant acceleration in 2 dimensions, I used a result from our 1 dimensional discussion - twice - once in the x dimension and once in the y dimension. It may not be obvious to you that it is legitimate to treat the x and y motion of a single particle as two instances of 1 dimensional motion. Think of it this way. Any acceleration in the x direction can only change the x direction velocity since the change in velocity is just the acceleration times Dt and Dt is a scalar. Likewise velocity in the x direction can only change x direction position for the same reason. So as long as the accelerations along each dimension are independent, so are the velocities and positions. We may consider systems where the accelerations in different dimensions are coupled in some way, but that is a complication we will save for later.

So for constant acceleration in 2 dimensions (or 3 dimensions for that matter) when the reference frame is chosen sensibily and there are no weird connections between accelerations in different dimensions, we can adopt the results of our 1 dimensional constant acceleration study directly. In particular let's look at a particle under the influence of the Earth's gravity. We have already established that gravity near the surface of the Earth provides a practically constant acceleration straight down of 9.8 m/s2 so one of our reference frame axes should lie in that direction. I will place the -y axis in the direction of the gravitational acceleration, consistent with common practice.

Fixing the y axis with gravity, vertically, locks in our 2 dimensional (plane) reference frame to a plane perpendicular to the Earth's surface. The other vector we need to accomodate in the plane is the initial velocity. So we orient the reference frame so as to contain the initial velocity vector. The x axis then is a horizontal line in the plane defined by the acceleration vector and the initial velocity vector. Now that we have built our reference frame, just place it so that the particle lies in the plane and we are ready to examine the particle's motion. As long as our assumption of constant (unchanging in either magnitude or direction) acceleration is true, the particle, wherever it may go, will not wander out of the reference frame plane.

projectile motion
At this point your physics text probably has lots of examples of what is called "projectile motion". They all come down to the application of the laws we worked out for 1 dimensional, constant acceleration, motion applied separately to the x and y dimensions of the reference frame. The key to these problems is to take the given information which usually boils down to an initial velocity vector and resolve the initial velocity into its x and y components. Those components then are the initial velocities to be used in the equations describing the x and y motion.

The Projectile Motion display automatically solves projectile motion problems for you once you sift the initial velocity vector out of the given information. You should recognize it as similar to the Parametric Plot display shown earlier. Play around with this display to build up your intuition about projectile motion.

In some physics texts you may find at this point some more kinematics, involving uniform circular motion and multiple reference frames. We will hold off on discussion of these topics. Uniform circular motion will be covered after we get to know something about the relationship between force and motion. Multiple reference frames will be covered in the section on the nature of space.

At this point there is a choice to be made about how to get at the laws of nature. One approach is to come in at a very profound level which involves the interaction of space and matter, and some sort of "least action" principle, following the trail blazed by Lagrange and Hamilton. The advantage of this way is that the laws of nature fall out logically from a single assumption. The disadvantage is that this requires a degree mathematical sophistication that would take us a long time to establish from the base we have built so far.

The other approach is to develop the laws of nature from the common sense which grows out of experience. It was Sir Isaac Newton who first expressed this common sense in formal terms. The advantage of this approach is that our intuition about nature is consistent with this development. The disadvantage is that more of the facts of dynamics are considered to be laws, requiring us to remember more and understand less. Also the connections among the various laws are not necessarily obvious.

We will take the Newtonian tack at this point. For those who are interested in the less intuitive and more elegant discussion there will be another course from M. Casco called Relativity. Also at the end of this course there is a section called the Nature of Space where we fill in some of the background behind Newton's laws of motion. If you are interested in this approach you might want to digress at this point and work through that section now. Otherwise you may continue on with the next section, taking Newton's laws to be the fundamental principles.

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