Displacement
Figure 1 These cyclists in Vietnam can be described by their position relative to buildings and a canal. Their motion can be described by their change in position, or displacement, in the frame of reference. (credit: Suzan Black, Fotopedia)
By the end of this section, you will be able to:
Position
In order to describe the motion of an object, you must first be able to describe its position—where it is at any particular time. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the position of the rocket with respect to the Earth as a whole, while a professor's position could be described in terms of where she is in relation to the nearby white board. (See Figure 2.3.) In other cases, we use reference frames that are not stationary but are in motion relative to the Earth. To describe the position of a person in an airplane, for example, we use the airplane, not the Earth, as the reference frame. (See Figure 1.)
If an object moves relative to a reference frame (for example, if a professor moves to the right relative to a white board or a passenger moves toward the rear of an airplane), then the object's position changes. This change in position is known as displacement. The word “displacement” implies that an object has moved, or has been displaced.
Displacement is the change in position of an object:
where \(\Delta x\) is displacement, \(x_f\) is the final position, and \(x_o\) is the initial position.
In this text the upper case Greek letter \(\Delta\) (delta) always means “change in” whatever quantity follows it; thus, \(\Delta x\) means change in position. Always solve for displacement by subtracting initial position \(x_o\) from final position \(x_f\).
Note that the SI unit for displacement is the meter (m) (see Physical Quantities and Units), but sometimes kilometers, miles, feet, and other units of length are used. Keep in mind that when units other than the meter are used in a problem, you may need to convert them into meters to complete the calculation.
Figure 2 A professor paces left and right while lecturing. Her position relative to the blackboard is given by \(x\). The \(+2.0\;m\) displacement of the professor relative to the blackboard is represented by an arrow pointing to the right.
Figure 3 A passenger moves from his seat to the back of the plane. His location relative to the airplane is given by \(x\). The \(-4\) m displacement of the passenger relative to the plane is represented by an arrow toward the rear of the plane. Notice that the arrow representing his displacement is twice as long as the arrow representing the displacement of the professor (he moves twice as far)
Note that displacement has a direction as well as a magnitude. The professor's displacement is 2.0 m to the right, and the airline passenger's displacement is 4.0 m toward the rear. In one-dimensional motion, direction can be specified with a plus or minus sign. When you begin a problem, you should select which direction is positive (usually that will be to the right or up, but you are free to select positive as being any direction). The professor's initial position is \(x_0=1.5m\) and her final position is \(x_f=3.5m\). Thus her displacement is
In this coordinate system, motion to the right is positive, whereas motion to the left is negative. Similarly, the airplane passenger's initial position is \(x_0=6.0m\) and his final position is \(x_f=2.0m\), so his displacement is
His displacement is negative because his motion is toward the rear of the plane, or in the \(negative\ x\) direction in our coordinate system.
Although displacement is described in terms of direction, distance is not. Distance is defined to be the magnitude or size of displacement between two positions. Note that the distance between two positions is not the same as the distance traveled between them. Distance traveled is the total length of the path traveled between two positions. Distance has no direction and, thus, no sign. For example, the distance the professor walks is 2.0 m. The distance the airplane passenger walks is 4.0 m.
It is important to note that the distance traveled, however, can be greater than the magnitude of the displacement (by magnitude, we mean just the size of the displacement without regard to its direction; that is, just a number with a unit). For example, the professor could pace back and forth many times, perhaps walking a distance of 150 m during a lecture, yet still end up only 2.0 m to the right of her starting point.
In this case her displacement would be +2.0 m, the magnitude of her displacement would be 2.0 m, but the distance she traveled would be 150 m. In kinematics we nearly always deal with displacement and magnitude of displacement, and almost never with distance traveled. One way to think about this is to assume you marked the start of the motion and the end of the motion. The displacement is simply the difference in the position of the two marks and is independent of the path taken in traveling between the two marks. The distance traveled, however, is the total length of the path taken between the two marks.
What is the difference between position and frame of reference?
Which statement best describes the total displacement for a round trip?
What is the total displacement for a round trip?
A boy walks from home 459 m East then 379 m North to reach his school. If he can take another way straight from home to school, what is his displacement?
Express your answer in meters.
The distance is the straight-line distance from starting point to finish in a specified direction.
\(c=\sqrt{a^2 + b^2}\)
\(c=\sqrt{(459\;m)^2 + (379\;m)^2}\)
Answer: 595.2495
Dina went to the market. If she walked 8.3 m from home to market, how much distance (in meters) does she walk back and forth?
\(d_t=2(8.3 \space m)\)
Answer: 16.6
Roy walks from home 2 m East then 3 m North to reach his school. If he can take another way straight from home to school, what would be his displacement?
Myla walks from home 2 m East then 3 m North to reach her school. One day, she took another way straight from home to school which it just took her to walk 3.6 m. This measurement is referred as what?
Roy walks from home 2 m East then 3 m North to reach his school. If he can take another way straight from home to school, which of the following formula is the most appropriate to use to determine his displacement?
The total displacement for a round trip is either negative or positive, depending on the direction of the motion.
Total displacement for a round trip is zero, regardless of the path taken or distance traveled.
A displacement is always a straight arrow directed from the ending point to the starting point.
We can define a _________ as a coordinate system which is often used to study the movement of a particle.
It measures the straight path from the initial position to the final position.
Select two statements that are true about distance.
In what direction of displacements \(\vec{C}\) and \(\vec{D}\) would the magnitude of the total displacement have a minimum value?
___________ refers to the physical length an object travels and is always positive.
Distance
In what direction of displacements \(\vec{C}\) and \(\vec{D}\) would the magnitude of the total displacement have a maximum value?
\(\vec{X}\) is equal to 15 m and \(\vec{Y}\) is 9 m. What would be the possible maximum value of the total displacement of \(\vec{X}\) and \(\vec{Y}\) ?
A displacement is always a straight arrow directed from the starting point to the ending point.
A fundamental tenet of physics is that information about an event can be gathered from a variety of reference frames. For example, imagine that you are a passenger walking toward the front of a bus. As you walk, your motion is observed by a fellow bus passenger and by an observer standing on the sidewalk.
Both the bus passenger and sidewalk observer will be able to collect information about you. They can determine how far you moved and how much time it took you to do so. However, while you moved at a consistent pace, both observers will get different results. To the passenger sitting on the bus, you moved forward at what one would consider a normal pace, something similar to how quickly you would walk outside on a sunny day.
To the sidewalk observer though, you will have moved much quicker. Because the bus is also moving forward, the distance you move forward against the sidewalk each second increases, and the sidewalk observer must conclude that you are moving at a greater pace.
To show that you understand this concept, you will need to create an event and think of a way to view this event from two different frames of reference. In order to ensure that the event is being observed simultaneously from both frames, you will need an assistant to help out. An example of a possible event is to have a friend ride on a skateboard while tossing a ball. How will your friend observe the ball toss, and how will those observations be different from your own?
Your task is to describe your event and the observations of your event from both frames of reference. Answer the following questions to demonstrate your understanding.
What is your event?
What object are both you and your assistant observing?
What do you see as the event takes place?
What does your assistant see as the event takes place?
How do your reference frames cause you and your assistant to have two different sets of observations?