[dropcap]G[/dropcap]Graphs have two axis on which two quantities can be represented and relation of the two quantities can drawn. We know that runs scored by the cricket team with overs are shown a bar graph on the television. The graphs are representation of events which can be used for comparisons etc. In mathematics, the line graphs help in solving linear equations.
In motion, distance,time,speed are the quantities which can be drawn on the graph.We will use line graphs between distance and time or speed and time.
1. Distance – Time graph
The change in the position of a moving body over a time interval can be represented using line graph between distance and time. The graph is plotted with time on x-axis and distance on y-axis. In case of uniform speed, the body travels equal distances over equal time intervals. In such case the distance covered is proportional to the time because it is increasing with time at uniform rate. Therefore, the graph is a straight line in which the distance is increasing with time at uniform rate.
We can use the distance-time graph to determine the speed. To do so, consider a small part AB of the graph as shown in fig. Draw a line parallel to x-axis from A and draw a line parallel to y-axis from B. These lines meet at C.The right angled triangle ABC has AC as time interval t2– t1 and BC as distance traveled during the time interval s2– s1. We can see from the graph that as the body moves from A to B on the graph, it covers distance s2– s1 during the time interval t2– t1 . This can be used for calculating the speed during the time interval.
If speed of the body is represented as ‘v’ then
In any distance time graph the slope of the graph provides speed .
If the y-axis shows displacement the above formula becomes for velocity because the displacement is uniform across all intervals. If the body is moving with non-uniform speed, then the distance time graph will not be a straight line. For a body falling due to gravity, the speed is not uniform and the graph is not a straight line. In this case the speed is increasing at a constant rate and the distance traveled in each subsequent second is more than the distance traveled during the previous second. Fig. below shows such a motion.
2. Velocity-Time graph
The velocity of an object moving along a straight path can be plotted with time as a line graph. In the graph, time is along x-axis and velocity is along y-axis. Following cases are for the graph –
(a) When object is moving with uniform velocity.
In this case the height of the graph will not change with time as shown in fig.
The product of velocity and time give displacement for the object moving with uniform speed. The area enclosed by velocity-time graph and the time axis will be equal to the displacement. To know the distance moved by the object between time t1 and time t2, draw the perpendiculars from t1 and t2 to the graph. The perpendiculars intersect the graph at A and B respectively. The area ABDC gives the distance covered by the object. So the distance covered s during the time interval t2– t1 is = AC x CD
(b) When the object is moving with uniform acceleration.
In this case the change in velocity for equal time intervals is equal. Therefore, the change in velocity is
proportional to the time period. If the acceleration is positive, the graph between velocity and time will be a straight with positive slope because slope is equal to the acceleration.Following table shows such a motion of a car in which the velocity is recorded at different instants.
When the velocity is plotted against time for this table, the velocity is a straight line which is having increasing trend therefore it has positive slope along time axis. The plot is shown in fig. below. The nature of the graph shows that velocity changes by equal amounts in equal intervals of time. Thus, for all uniformly accelerated motion, the velocity time graph is a straight line.
You can also determine the distance moved by the car from from its velocity-time graph. The area under the velocity-time graph gives the distance moved by the car in a given interval of time.
The distance moved by the car during time interval t1 and t2 is the area under the velocity-time plot between t1 and t2. This is area of ABCED. The area of ABCED is made up of rectangle ABCD and a triangle CDE.
The area of a triangle is (1/2 x base x height).
In the fig. below, the velocity-time graph shows that the velocity is decreasing uniformly over the uniform time intervals.
The velocity-time graph in fig. shows that the acceleration of the object is non-uniform.
#Exercise – What is the nature of the distance-time graph for uniform and non-uniform motion?
Solution –
(i) The distance-time graph is a straight line for uniform motion because the change in distance is uniform for uniform time intervals.
(ii) The distance-time graph is not a straight line for non-uniform motion.
#Exercise – What can you say about the motion of an object whose distance-time graph is a straight line parallel to the time axis?
Solution – If the distance-time graph is a straight line parallel to the time axis, it means that the distance is not changing with increase in time. That means the distance of the object is constant and the object is not moving.
#Exercise – What can you say about the motion of an object if its speed-time graph is a straight line parallel to the time axis?
Solution – If the speed-time graph is a straight line parallel to the time axis, it means that the speed is not changing with the increase in time. That means the speed of the object is constant . That means the motion is uniform motion. The acceleration of the object is zero.
#Exercise – What is the quantity which is measured by the area below the velocity-time graph?
Solution – The area provides the distance traveled by the object.
Exercise – What is the quantity which is measured by the slope of the distance-time graph?
Solution – The slope of distance-time graph provides the speed at the point where the slope is
measured.
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Equations of motion by graphical method