Position, Displacement, and Distance

Section 2.3 Position, Displacement, and Distance

Motion of an object moves the object from one place to another. We track the motion using distinct concepts of position, displacement, and distance. While position refers to the object at one particular instant of time, displacement and distance refer to what happens over an interval of time as explained below.

Subsection 2.3.1 Position

Position of an object refers to its location in space at a particular instant. Clearly, location will be a point if the object is point-like such as an electron. But, what do you do if you have an extended object such as a truck or Earth? We will see in a later chapter that, for translational motions, it is sufficient to treat even an extended object by a point placed at a special point, called the center of mass. By position of an object we will mean this special point without further elaboration here.

For a motion occuring on a straight path, we will often choose \(x\) axis to coincide with the path of motion. In that case \(x \) coordinate of the object will be its position. For vertical motions, we often use \(y\) coordinate for the up and down straight path; in that case, it is the \(y\) coordinate that will give position.

Most of our examples will be motion in a plane, either horizontal plane, vertical plane, or slanted plane. For these types of motion, it is customary to use \(xy\)-plane for the plane. We, then use either the Cartesian coordinates \(x\text{,}\) \(y\) or polar coordinates, \(r \text{,}\) \(\theta \) for the motion. Polar coordinates are particularly useful when the motion is on a circle or an ellipse.

For arbitrary motion in a three-dimensional space we can use all Cartesian coordinates, \(x,\ y,\ z \text{,}\) for the position. Other coordinate systems, such as spherical and cylindrical coordinates, are also used in particular situations. We probably will not have many examples of three-dimensional motion.

Subsection 2.3.2 Displacement

Displacement is the change in position during an interval. Displacements along coordinates are denoted by \(\Delta x, \Delta y, \) and \(\Delta z\text{.}\) Read them as "delta \(x\)", "delta \(y\)", "delta \(z\)".

Suppose, an object moves on a straight line. With \(x\) axis along the motion, let position at initial instant \(t_i \) be \(x = x_i\) and at final instant \(t_f \) be \(x = x_f\) . Then, displacement during the interval \(t_i \) to \(t_f \) will be

\begin{equation} \Delta x = x_f - x_i.\tag{2.1} \end{equation}

Clearly, if \(x_f \gt x_i\text{,}\) displacement will be positive and if \(x_f \lt x_i\text{,}\) it will be negative. Therefore, the sign of \(\Delta x\) has the information about the direction of motion on the \(x\) axis: positive means motion towards the \(x=+\infty\text{,}\) negative means towards \(x=-\infty\text{.}\)

Figure 2.13. A to B has \(\Delta x \gt 0\) and B to C has \(\Delta x \lt 0\text{.}\)

Subsection 2.3.3 Distance

Distance traveled during an interval, to be denoted by \(d \text{,}\) refers to the length of the entire path of the motion, not just the difference between initial and final positions. The distance is not the same thing as displacement. In Figure 2.13, the displacement of motion A-to-B-to-C is \(x_C-x_A\text{,}\) but distance travelled is sum of the distances AB and BC.

\begin{equation*} d = \text{AB} + \text{BC} = |x_B - x_A|+|x_C - x_B|,\ \ \ \text{but},\ \ \ \Delta x = x_\text{C} - x_\text{A}, \end{equation*}

where \(|x_B - x_A|\) is the absolute value of \(x_B - x_A\) and similarly for BC.

As another example, consider going around a circle, returning to the same place. In this case, the displacment is zero because final position is same as the initial position, but the distance will be the circumference of the circle. The difference between displacement and distance is further illustrated in Example 2.14.

Example 2.14. Difference between Displacement and Distance.

A boy runs on a straight track, which is taken to be the \(x \) axis. As shown in Figure 2.15, the boy is at \(x = 5\text{ m} \) mark at \(t = 2 \text{ sec} \text{.}\) He runs to \(x = 50\text{ m}\) mark, reaching there at \(t=10\text{ sec}\text{.}\) He then turns around and runs back to \(x = 20\text{ m}\) mark arriving there at \(t = 20 \text{ sec}\text{.}\)

(a) What is his displacement during the \(t = 2 \text{ sec} \) to \(t = 20 \text{ sec} \) interval?

(b) How much distance did he travel during the \(t = 2 \text{ sec} \) to \(t = 20 \text{ sec} \) interval?

(d) How much distance did he travel during the \(t = 2 \text{ sec} \) to \(t = 10 \text{ sec} \) interval?

(e) What is his displacement during the \(t = 10 \text{ sec} \) to \(t = 20 \text{ sec} \) interval?

(f) How much distance did he travel during the \(t = 10 \text{ sec} \) to \(t = 20 \text{ sec} \) interval?

Answer.

(a) \(\Delta x = 15\text{ m}\text{,}\) (b) \(d = 75\text{ m}\text{,}\) (c) \(\Delta x = 45\text{ m}\text{,}\) (d) \(d = 45\text{ m}\text{,}\) (e) \(\Delta x = -30\text{ m}\text{,}\) (f) \(d = 30\text{ m}\text{,}\)

Solution 1. a

The displacement on \(x \) axis is the change in the \(x \) coordinates.

\begin{equation*} \Delta x = 20\text{ m} - 5\text{ m} = 15\text{ m}. \end{equation*}

Solution 2. b

The distance traveled on path must take into account all the distance traveled. Here, the boy first ran \(50-5 = 45 \text{ m}\) and then ran another \(50 - 20 = 30\text{ m} \) on the return path. We need to add them.

\begin{equation*} d = 45 \text{ m} + 30\text{ m} = 75\text{ m}. \end{equation*}

Solution 3. c

The displacement on \(x \) axis is the change in the \(x \) coordinates.

\begin{equation*} \Delta x = 50\text{ m} - 5\text{ m} = 45\text{ m}. \end{equation*}

Solution 4. d

(d) Since there is no turn in motion direction during this interval, the distance is equal to the magnitude of displacement.

\begin{equation*} d = 45\text{ m}. \end{equation*}

Solution 5. e

The displacement on \(x \) axis is the change in the \(x \) coordinates.

\begin{equation*} \Delta x = 20\text{ m} - 50\text{ m} = -30\text{ m}. \end{equation*}

Solution 6. f

Since there is no turn in motion direction during this interval, the distance is equal to the magnitude of displacement, stripping away the negative sign.

\begin{equation*} d = 30\text{ m}. \end{equation*}

Exercises 2.3.4 Exercises

1. Practice with a Friend.

A bottle rocket (firework) is shot vertically up from a platform (point A) that is \(10\text{ m}\) above ground. The rocket flies to a height of \(50\text{ m}\) (point B) before turning back and falling to the ground (point C). Use \(y\) axis for displacements. What are the displacements and distances tavelled for the following motions: (a) A to B, (b) B to C, and (c) A-to-B-toC?

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