Motion in 1-dimension

The work of a curious fellow
This thing is all long and no wide...

We will begin this lesson in mechanics by throwing out some of the complicating issues which clutter the general picture. Then when we have a solid understanding of the basics we can add these discarded factors back into our thinking.

For the time being we will describe the motion we are studying in terms of space and time without explicitly considering the agents that cause the motion. This simplification is so common that a name has been given to this branch of mechanics. It is called kinematics . One of the fundamental concepts in kinematics is displacement which is the difference in position of an object at two different times. In this lesson we will combine the displacement idea with the rate of change notion to develop velocity and acceleration . Then we will use the concepts of displacement, velocity and acceleration to study the motion of objects.

There are three types of motion that we will study in mechanics. These are translation, rotation and vibration. Translation is motion along some path from one place to another, like a car moving down a highway. Rotation is motion around some axis, like the Earth's daily motion. Vibration is a back and forth motion like the pendulum of a clock. For the purposes of this lesson we will not consider rotation, limiting ourselves to motion in a straight line. This simplification eliminates the need to use vectors to measure the quantities describing the motion. The straight-line motion is the reason we use 1 Dimension in the title of this lesson. Also by placing this restriction on the motion we will study, we only need to consider objects that are particles , meaning that their size does not enter into our consideration.

Since we are going to be interested in motion in a straight line, we can limit our reference frame to a single real number line and let each number on the line represent a unique position of the particle. Let's call this number line the x axis, so that x stands for particle position. The motion of a particle is completely known if for every time, t, we know its position, x. A convenient way to display the motion of a particle then is to plot its position vs. time on a two dimensional graph as shown in the Motion Plot display.

Position vs. Time
Position and Velocity vs. time

In our discussion so far we have made a distinction between velocity and speed, where speed is the absolute value of the rate of change of displacement with respect to time and velocity includes both a magnitude and direction. In the 1 dimensional case we have here, the only direction information required is whether the motion is towards increasing position numbers (positive), or towards decreasing position numbers (negative). Still, some direction is given by the algebraic sign so we call the rate of change of displacement with respect to time, velocity.

Remember from our rate of change discussion that the rate of change of one variable with respect to another is the ratio of the change in the dependent variable to the corresponding change in the independent variable. What that means in our case is that the rate of change of position with respect to time between P1 and P2 is (x2 - x1) / (t2 - t1). So Dx = x2 - x1 and D t = t2 - t1 In this special case where x represents position, Dx is the displacement.

Because we knew the position and time at two instances, we were able to calculate a velocity v = Dx / Dt . To indicate that this velocity is only the average velocity between position x1 and x2, we symbolize it with a bar over the v, like this average velocity, so average velocity = Dx / Dt .

In general if we use the subscript i to denote initial conditions and the subscript f to denote final conditions, then Dx = xf - xi and Dt = tf - ti , so average velocity = Dx / Dt = (xf - xi) / (tf - ti) .

devilish hardBe careful in the order in which you subtract one value from another to get the D of a variable. If you are inconsistent, the sign of the velocity will come out wrong. Most folks subtract the initial value from the final value. This is a convention that helps us keep our signs straight. It is a good practice to use, even if other schemes might work.

Let's calculate the average velocity between t1 and t2 from the data we were given. That gives us average velocity = (13-3) / (16-1) = 10/15 = .6667 m/s (meters per second) , to 4 significant figures. You may have become accustomed to answers that come out even. In physics, whatever happens... happens.

Now we know from our work with rates of change that the instantaneous rate of change of position with respect to time is the slope of the position vs. time plot, so instantaneous velocity is the slope of the position vs. time plot at the instant in time we are considering. In the Position & Velocity vs. Time display we have measured the slope of the position vs. time plot at many points along the time axis and plotted the results.

Acceleration is related to velocity in exactly the same way that velocity is related to position. Average acceleration average acceleration is defined as the change in velocity Dv divided by the elapsed time D>t so average acceleration = Dv / Dt = (vf - vi) / (tf - ti)

Instantaneous acceleration at ti is defined as the limit of (vf - vi) / (tf - ti) as tf approaches ti, the same process by which we found instantaneous velocity from the average velocity. Remember that in terms of calculus velocity is the first derivative of position with respect to time and acceleration is the second derivative of position with respect to time. In the Position, Velocity & Acceleration vs. Time display we illustrate the relationship among position, velocity and acceleration for our example, as functions of time.

The motion we have been studying illustrates a principle that we will see again, that of superposition of motions. Since our particle ends up at a place some distance from its starting point, there is a translational component of the motion. But it does not get there without some back and forth movement, so there is also a vibrational motion involved. The Actual Motion Plot display shows the actual motion of the particle on the plot we have been working with.

Are there any questions?

Study the position, velocity and acceleration plots to picture the motion as a combination of pure vibration and pure translation. The actual equation of the motion in this example is x=3+.625*(t-1)+sin(t-1) . Perhaps that information will help you to identify the translational part and the vibrational part.

From here we will move on to study motion in one dimension where the acceleration is constant.

position velocity and acceleration vs. time
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