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2. Motion Along a Straight Line

By Michael Tanczos | Published Aug 06 1999 07:50 AM in Math and Physics

velocity acceleration average formula instantaneous time displacement position object
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Motion Along a Straight Line

Everything in our world moves. Even stationary things such as your house move in relation to the sun, or to other stars, or .. The classification and comparison of motion is called kinematics and is quite challenging. Before we get too far, I'd like to define a few things.
  • The motion we will be covering in this section will be in a straight line only. This motion can be horizontal, vertical (falling), or even slanted, but it will always be in a straight line.
  • We will not specify the cause of this motion in this section. We will assume specific properties an object will have as it is in motion.
  • An object will be defined as a particle (such as an electron) or an object with particle-like properties where all parts of the particle move uniformly without any rotation. Ex. A book falling from a certain height would qualify, but only if you did not apply any sort of spin to that book.
Knowing that, here is what we will cover:
  • Position and Displacement
  • Average Velocity and Average Speed
  • Instantaneous Velocity and Speed
  • Acceleration
  • Constant Acceleration
  • Free-fall Acceleration
Position and Displacement

When we try to locate an object in physics, we will usually relate it to some sort of reference point. This point is usually the origin, the zero point of an axis (x,y, or z) Furthermore, we provide a numeric representation for an object's location by placing it either in the positive direction (+) or negative direction (-) in relation to that reference point.

For example, lets put a particle at x = 10 meters. The particle would then be located at 10 meters from the origin on the x-axis. As you may have noticed, position is marked by some unit of length. The typical unit we will use is meters. Now, on to our first formula!

The difference between an objects starting position and final position after it moves is known as displacement. Displacement is found with the simple formula:

Formula 1.01 -- Displacement

Attached Image: form101.gif

Where:

Displacement is the change in X position, or delta X. Delta is a Greek Symbol which looks similar to a triangle.

X2 = Destination position

X1 = Starting position

If we ignore the sign that we get from Formula 1.01, we are left with the magnitude of the displacement.

Displacement is what is known as a vector quantity. A vector quantity has both a direction and a magnitude. We will learn more about vectors in later sections.

Average Velocity and Average Speed

When we describe the actual motion of an object, we tend to describe it's position x as related to time. Several quantities are associated with the motion of an object over a specific phase of time. One of these quantities is known as Average Velocity. Average Velocity is the ratio of displacement to change in time. Which brings us to our next formula.

Formula 1.02 -- Average Velocity

Average Velocity = Displacement / Change in time

Or

Attached Image: form102.gif

If we were to create a velocity vs time graph for a constant velocity, the slope of any point on the resulting line we will get is the average velocity.

The second thing we will take a look at is Average Speed. Please note that speed is not the same thing as velocity. Average Speed is not a vector quantity and has only a magnitude. It is calculated very much like average velocity except that instead of calculating the total displacement, we calculate the total distance a particle travels during an interval of time. So speed can be found with the formula:

Formula 1.03 -- Average Speed

Attached Image: form103.gif

Note:

The little line above the 's' denotes that value as being 'average'.

Instantaneous Velocity and Speed

You have seen how to calculate an average velocity of a particular object, but there are times when we would like to calculate exactly how "fast" an object is moving at a single instant in time.

Instantaneous velocity is obtained from average velocity by shrinking the difference in times (delta t) closer and closer to zero. We eventually reach a limiting value which is known as instantaneous velocity. If we would create a graph of velocity vs. time for an object, we could derive the instantaneous velocity by taking the slope of a line tangent to any point on the curve (of that velocity). However, we won't be calculating Instantaneous Velocity through a graph if we want to use it inside our programs. So, we will need some basic Calculus to find this velocity. Here is the formula for Instantaneous Velocity:

Attached Image: form104.gif

In plain English, the formula can be read as "The limit of the displacement divided by the change in time as the change in time approaches zero" or "the derivative of x with respect to t"

Calculating instantaneous speed is simply taking the solution to the above formula and stripping away it's direction and keeping only the magnitude of the instantaneous velocity.

Acceleration

When an object 's velocity changes over an interval of time, it undergoes some form of acceleration. Similar to velocity, there are two forms of Acceleration. Average acceleration, and Instantaneous acceleration.

The formula for Average acceleration is as follows:

Formula 1.05 -- Average Acceleration

Attached Image: form105.gif

The formula for Instantaneous acceleration:

Formula 1.06 -- Instantaneous Acceleration

Attached Image: form106.gif

Instantaneous acceleration at any point in time is the rate in which velocity changes at that instant. We can also relate acceleration to displacement by using a form of the above formula for instantaneous acceleration. In other words, we can again find instantaneous acceleration by taking the second derivative of an object's position x(t) with respect to time.

Commonly, our units for acceleration is meter per second per second, or m/s^2. Acceleration is also a vector quantity like velocity and can be both negative and positive. A negative acceleration is still acceleration, but is sometimes referred to as deceleration.

Large units of accelerations are often expressed in "g" units, where

1g = 9.8 m/s^2 (g unit)

Constant Acceleration

Constant acceleration is a special case. When dealing with a constant acceleration we are also dealing with a velocity that is increasing at a constant rate. So, if we graph acceleration vs time we are left with a line with a slope equal to zero.

Please note that the following equations are for constant acceleration only!

Formula 1.07 -- Velocity at a Certain Instant

Attached Image: form107.gif

Formula 1.08 -- Displacement

Attached Image: form108.gif


On to the next section..





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