LINEAR MOTION AND GRAPHS

Dr. Cahit Erkal

 

We will study two types of graphs: displacement-time and velocity-time.We will then learn how to obtain a velocity versus time graph from the displacement-time graph (also the reverse process).This presentation can be adapted to any grade level between 5 and 12 with some modifications.It presents the most fundamental concept of physics, the linear motion with minimal mathematics, while providing a test-ground for the skills developed in mathematics classes.The goal of this exercise is to introduce physics early on, with the hope that, it will reinforce some of the mathematical concepts learned--while encouraging the teachers to include at least one unit of description of motion.

 

Arithmetic skills practiced in this exercise:

Number line, concept of direction on the number line (coordinate frame is the fancy name), slope, finding areas of such geometric shapes like triangle or rectangle, graphing, tables (charts).

 

Displacement:

 

Position of an object (what you would read from the meter stick next to which a toy car is in motion on a straight line), as it moves, changes with time.This position change relative to an origin (a reference point) is called DISPLACEMENT.Displacement is a vector quantity (means it has duration as well as size), whereas distance covered is a scalar quantity (it has no direction, but has size).

 

Examples†††††††††††††††††††††††††† B

Car-1

 
††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† †††††††† .†††† ††††A

1)†††††††††††††††††††† †††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ††††††††††††††††††††††††††††††††††††††††††††

 

 

 

 

††††††††††††††††††††††††††††††††††† A denotes the displacement of Car-1

††††††††††††††††††††††††††††††††††† B denotes the displacement of Car-2

 

†† Both cars are at the same distance from origin, however they have†††††††††††††††††††††††††††††† different displacements.

 

 

2) Letís consider a coordinate frame (two meter sticks placed mutually perpendicular), which measures the displacements in horizontal (x) as well as vertical (y) directions.

 

 

 

 

 

 

 

 


†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† 4††††††† ††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

A denotes the position of the car at this††††††††††††††††††††††††††††† 3†††††††††††††††††††††††† A††††††††††††

particular time; B denotes the position

of the car at a later time.††††††††††††††††††††††††††††††††††††††††††††††††††††† 2††††††††††††††††††††††††† B

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ††††††††††† 1†††††††††††††††††

x (m)

 
††††††††††† x,y

††††††† A: (2,3)††† B: (4,2)†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† †† `1††† 2††† 3††† 4

 

During the time the carís displacement changed fromA to B, its x-displacement changed from 2 to 4, its y-displacement changed from 3 to 2.

 

 

 

Graphs

 

Slope of a straight line = Rise/Run

 

 

You can calculate the slope by choosing any

two points on the line [such as A (X1,Y1) and

X2-X1=Run

 

 

X1†††††††††††††††††††††† X2

 
B (X2, Y2)].

 

X

 
 

 

 

 


††††††††††††††††††††††††††††††††††††††††††††††† y2-y1

††††††††††††††††††††††† Slope =††

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† x2-x1††††††††††††††††††††††††

 

 

 

 

 

 

 

Note that the application of slope in describing the velocity is accomplished by taking the horizontal axis as time(t) and the vertical axis(y) as displacement x.

 

 

††††††††††††††††††††††††††††††††††††††††††††††† †† x2-x1

†††††††††††††††††††††† Velocity†† V =††††† ††††††††††††††††††††††††††††††††velocity is the slope of x-t graph

††††††††††††††††††††††††††††††††††††††††††††††† ††† t2-t1

 

 

 

 

Letís record the displacements of a toy car as it moves in one dimension [it can go the positive x-direction(to the right) or in the negative x-direction (to the left).

 

 

Time (t)††††††††††††††††††††††† Displacement (cm)

††† 0††††††††††††††††††††††††††††† †††††††††† 0

††† 5††††††††††††††††††††††††††††† †††††††† 4.1

10††††††††††††††††††††††††††††† †††††††† 7.9

15††††††††††††††††††††††††††††† †††††† 12.1

20††††††††††††††††††††††††††††† †††††† 16.0

25††††††††††††††††††††††††††††† †††††† 16.0

30††††††††††††††††††††††††††††† †††††† 16.0

35††††††††††††††††††††††††††††† †††††† 18.0

40††††††††††††††††††††††††††††† †††††† 20.1

45††††††††††††††††††††††††††††† †††††† 21.9

50††††††††††††††††††††††††††††† †††††† 24.0

55††††††††††††††††††††††††††††† †††† ††22.1

60††††††††††††††††††††††††††††† †††††† 20.0

70††††††††††††††††††††††††††††† †††††††††† 0

 

 

 

††††††††††††††††††††††††††††††††††† †††††††††

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Note that the motion on x-t graph is described by a set of straight lines.We analyze the motion by breaking the whole graph into segments, each of which describes the motion between the break point (20 sec, 30 sec, 50 sec, 60 sec).Realistically speaking, the lines would smoothly join to each other in an actual experiment.

 

 

Velocity = (x2-x1)/(t2-t1)

 

 

t2-t1 (s)†††††††††††††††††††††† displacement (cm)†††††††††††††††††††† velocity (cm/s)

††† 20-0†††††††††††††††††††††††† †††† ††16-0††††††††††††††††††††††††††††††††† 16/20 = 0.8

30-20=10††††††††††††††††††† †††† 16-16 = 0†††††††††††††††††††††††††† 0/10 = 0

50-30=20††††††††††††††††††† †††† 24-16 = 8†††††††††††††††††††††††††† 8/20 = 0.4

60-50=10††††††††††††††††††† ††† 20-24 = -4†††††††††††††††††††††††††† -4/10 = -0.4

 

Now we are ready to plot (or graph) v Ė t.

 

 

 

 

 

 

 

 

 

 

 

 

 


††††††††††††††††††††††††††††††

At the end points (20,30,50,60 seconds) velocity is not defined!Because the lines on x-t graph break , slope is undefined.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Letís plot the two graphs together.We can better see how the change in the displacement reflects on the v-t graph.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Note:

 

 

 

 

 

 

 

 

Velocity-time graph (v-t graph):

 

V-T graph displays the variation of an objectís velocity as a function of time.The slope of the v-t graph is acceleration a.Acceleration is the rate of change of velocity.Itís unit is cm/s2 or m/s2.We will study only constant accelerated motion.

 

Examples

 

1)

†††††††††††††††††††††††††††† 3-2

 

†† 40-20

 
††††††††††† v(cm/s)5††††††††††

††††††††††††††††††††††† 4†††††††††

††††††††††††††††††††††††† 3

 

††††††††††††††††††††††††† 2

††††††††††††††††††††††† 1†††††††††††††††††††††

 

††††††††††††††††††††††††††††††††††† 10††††† 20†††† 30†††† 40

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† t(s)††††††††††††††††††††††††††††††††††††††††††

 

slope =

 

††††††††††† This means that velocity of the object changes by 0.05 cm/s every second.

 

 

 

2)

 

v (cm/s)

 

5

 

3

 

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† †† 20††††††† 30 t (s)

 

††††††††††††††† ††††††††† 10††††††††††††† 15††††††††† ††

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† †††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

 

-5

 
†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

 

 

 

 

 

 

Area under v-t graph as displacement

 

Letís assume v-t graph is given and ask the question ďwhat is the displacement from time t1, to time t2

 

 

 

 

v(cm/s)

 

 

 

††††††††††††††††††††††† 10

 


††††††††††††††††††††††† 7

 


†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† t(s)

††††††††††††††††††††††††††††††††††† †††††† 10†††††††††††††† 20†††††††††††††† 30

 


†††††††††††††††††††††† -8

 

 

 

 

This is a negative area; height of the rectangle is Ė8 , width

is 10 seconds.It represents a negative displacement, which means that the object is now displacing toward the origin.Its direction of motion is opposite to its initial displacement.

 

Area of this rectangle is the displacement from t = 0 to t = 10s

 
 

 

 

 

 

 

 


time(s)

Displacement (cm)

0-10

area = 10x10 = 100

10-20

area = -8x(20-10) = -80

20-30

area = 7(30-20) = 70

 

Total distance covered = 100+80+70 = 250 cm

Total Displacement = 100 +(-80) + 70 = 90 cm, this means that object is 90 cm away from the origin, in the positive direction.

 

 

Reverse Problem: Given the v-t graph, sketch the motion:

In thispart of this exercise, you can ask the students to simulate the motion by moving forward and backward along a straight line, or use a remote controlled toy car alongside a meter stick.The idea is to be able to identify the information provided on the graph with the real life situation.If students can see how motion can be analyzed using real life examples, theyprocess the information more effectively.

 

We can sketch the approximate motion of the object for which the v-t graph is known.Take the example we just worked out:

 

0-10 s : object has initial velocity of 10 cm/s and displaces 100 cm in the first 10 sec.

 

10-20s: at t=10 sec it reverses (its velocity becomes negative) direction and covers 80 cm in the negative x direction which maintaining 8 cm/s velocity in the negative direction.

 

20-30 s: at t = 20 s it reverses direction and moves 70 cm in the positive x direction.

 

We can now sketch this as movements either along the horizontal number line:

 

 

 

 

†††††††††††