Concept of Position, distance, displacement, speed, and velocity

Position, distance, displacement, speed, velocity

• Distance time graph


At the end of this lesson, the students should be able to:
• Distinguish between distance and displacement in a translational motion
• Distinguish between speed and velocity
• Plot a distance-time graph and deduce the speed of motion from the gradient or slope of the graph
• Determine speed and velocity when simple problems are set involving distance or displacement and time

Content:

Concept of Position

The position of an object or a point is the location of that object or point in space with respect to a reference point.

The position of a point in a plane can be located by choosing axes, which intersect at right angles to each other. The vertical reference axis is called the y-axis or dependent axis while the horizontal axis is called the x-axis or independent axis. The point where they intersect is called the origin which is labelled (0, 0). The choice of the axes described above is called the Cartesian or rectangular co-ordinate. x is the distance of the point P from the origin along x-axis or horizontal while y is the distance of the point P from the origin along the y-axis or vertical.

How to locate the position of a point in a plane

Points in a 2-Dimensional Cartesian plane

The position of a point P (x, y) in the x – y plane can be found by moving x-units along the x-axis and y-units along the y-axis. Point A (4, 2) can be located by moving 4 units along the x-axis and 2 units along the y-axis.

If a particle moves on a line, its position is determined by measuring how far it is from the origin of the line. All points to the right of the origin are assigned positive values while all distances to the left of the origin are negative. The point P is 3 units to the right of the origin O and is positive.

The three-dimensional (3-D) cartesian plane can also be used to locate the position of points in space. The coordinate of a point, P in 3-D is generally of the form P(x, y, z).

A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the axes) that go through a common point (the origin), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes.

Evaluation:

Locate the following points on the Cartesian plane:

1. A(2, -1, 0) 

2. B(2, -3, 4)

3. C(2, 1)

4. D(0, -5)

5. E(-2, -3, -6)

Concept of Distance and Bearing

A bearing can be defined as the clockwise angular movement between two distant places. 

Types of bearing 

1. Three figure bearing e.g. 000°, 006°, 095°, 360° etc.

2. Compass bearing e.g. N60°W, N70°E, S45°W, S50°E, 46° North of West, 89° Easterly etc.

Rules for Solving Bearing and Distances. 

1. Taking reading in bearing starts from the North Pole in a clockwise direction and ends also at the North pole.

2. All angles formed while taking a reading in the bearing is equal to 360 degrees.

3. All questions in bearing leads to the formation of a triangle.

Concept of Distance

The distance covered by a body in moving from one place to another is the total length of the journey of the body.

Distance is the length of the line that joins two points together. The S.I unit of distance is meter (or metre), m. Distance is a numerical measurement of how far apart objects or points are. That is distance is only concern about the magnitude or amount of the length of a journey with no concern for the direction follows, thus it is a scalar quantity.

	=	distance
	=	coordinates of the first point
	=	coordinates of the second point

In 3-D, 

d = √[(x₂ - x₁)² + (y₂ - y₁)² +(z₂ - z₁)²]

where:
d is the distance between two points,
(x₁, y₁, z₁) are the coordinates of the first point,
(x₂, y₂, z₂) are the coordinates of the second point.

Solved Example:

1. Find the distance between two points: (8,8) and (-8,7)

Solution

x1 = 8

y1 = 8

x2 = -8

y2 = 7

Recall, d = √[(x₂ - x₁)² + (y₂ - y₁)²]

therefore: d = √[(-8 - 8)² + (7 - 8)²] 

d = √[(-16)² + (-1)²] 

= √[256 + 1] 

= √[257] 

= 16.03 units

2.

Three-Dimensional Distance Calculator

Evaluation:
Find the distance between the following points:
1. A(-1, 3) and B(-7, 5)
2. C(-2,3,4) and D(-1, 2, 3)
3. D(-1,0,4) and A(-2, -9, 0)
4. X(-2, 3, -5) and Y(1, 9, 5)
5. P(2, -9) and Q(8, 0)

Concept of displacement

Displacement is the distance travelled by a body or particle in a specified direction. It is the combination of the numerical value of the length between two points and the direction from the starting point to a particular destination. That is, displacement is the shortest distance between two points. Displacement can be positive, negative or zero.

So that, Displacement = distance + direction. And because a vector is a quantity with both magnitude and direction, then displacement is an example of vector quantities.

For example:

Total distance travelled in the diagram above is 7m but what is the displacement?

To answer this question, the displacement will be calculated using the Pythagoras rule, to determine the length of the line joining the start to the finish point. And that is: 

displacement = √[4² + 3²] = √[16 + 9] = √25 = 5m

So, it obvious that the total length of the journey, distance > the displacement of the body.

Also, consider the journey described by the diagram below:

 

Where Point A is the starting point and destination of the journey.

Evaluation:

1. Find the displacement and distance moved by the particle from point A through B to its destination, C in the diagram below.

2.  Find the total distance moved and the displacement from A to F in the diagram below. The graph scale is 1 unit on both axes.

Hint:

Concept of Speed

This is the rate of change of distance with time. It is the measure of how fast or slow a body moves. Or, it is the rate at which a body changes its position without considering direction from a  given frame of reference with respect to time. Thus, the measure of distance is always positive and it is a scalar quantity. The S.I unit of distance is meter, m.

The Odometer in cars measure distance.

Average speed, vavg:

This is the total distance, d travelled by a body per unit total time taken, t. That is:

vavg = dT/t

Also average speed, vavg = (v1 + v2)/2

Uniform or constant Speed, u:

This is the rate of change of distance, ∆d per unit equal change in time, ∆t i.e. 

u = ∆d/∆t

Uniform Speed: A body is said to be moving with uniform speed if it covers equal distances in equal intervals of time.

Variable speed: A body is said to be moving with variable speed if it covers unequal distances in equal intervals of time.

Uniform speed represents the slope of a distance-time graph.

Evaluation

1. A football field is about 100 m long. If it takes a person 20 seconds to run its length, how fast (what speed) were they running?

2. The pitcher’s mound in baseball is 85 m from the plate. It takes 4 seconds for a pitch to reach the plate. How fast is the pitch?

3. If you drive at 100 km/hr for 6 hours, how far will you go?

4. If you run a t 12 m/s for 15 minutes, how far will you go?

5. Every summer I drive to Pennsylvania. It is 895 km to get there. If I average 100 km/hr, how much time will I spend driving?

6. A bullet travels at 850 m/s. How long will it take a bullet to go 1 km?

7. Every winter I fly home to Chicago. It takes 3 hours. What is my average speed?

8. The fastest train in the world moves at 500 km/hr. How far will it go in 3 hours?

9. How long will it take light moving at 300,000 km/s to reach us from the sun? The

Concept of velocity

Velocity is the rate of change of displacement with time. It is a vector quantity and has the same S.I. unit of metre with speed. 

Mathematically, velocity (also symbolized as v) = displacement, s/time, t.

Average velocity, vavg

This is total displacement, sT of a body per unit total time taken, t.

So that, vavg = sT/t.

Uniform or constant velocity, u.

This is the rate of change of velocity per unit equal change in time, no matter how small the time change may be. That is, a body is said to be moving with uniform velocity if it covers equal displacement in equal intervals of time.

Uniform velocity represents the slope of a straight line displacetime-time graph.