Distance-Time Table and Distance-Time Graph

Mr.X is traveling from terminus A to terminus B in a bus and records his observation

 

Observations

Distance in km	0	10	20	30	40	50	60
Time	10.00am	10.15am	10.30am	10.45am	11.00am	11.15am	11.30am

The above table tells us that the bus is covering equal distances in equal intervals of time i.e., the bus is moving with uniform speed. If the bus continues to move with uniform speed then we can calculate the distance covered by the bus at any time.

Consider an object moving with uniform speed v from its initial position x_i to final position x_f in time t.



The equation (1) gives the relation between distance, time and average speed. This relation can be used for making distance-time tables and also to determine the position of any moving object at any given time.
But it is a long and tedious process especially when we have to determine the position after a long time or when we have to compare the motion of two objects. In such situations we can make use of graphs like distance-time graph.
A distance-time graph is a line graph showing the variation of distance with time.
In a distance-time graph, time is taken along x-axis and distance along y-axis.
Let us now plot a distance-time graph for the above example.

Distance-time graph for uniform motion

• Take time along x-axis and distance along y-axis.

• Analyze the given data and make a proper choice of scale for time and distance.

• Plot the points.

• Consider any two points (A, B) on the straight line graph.

• Draw perpendiculars from A and B to x and y axes.

• Join A to C to get a right angled triangle ACB.

Calculation:

• Write the title and scale chosen for the graph.

• Consider another two points P and Q on the graph and construct a right-angled triangle PRQ.

The distance-time graph is a straight line showing that the motion is uniform. Thus from the distance-time graph (S-t) we calculate the speed.

The ratio BC/AC is the slope of the graph. Hence, the slope of the distance-time graph gives the speed of the moving object.
In the second case also we get the speed as 0.66 km/min showing that the speed is uniform.
Now, let us see the nature of distance-time graph for a non-uniform motion. The following table gives the distance covered by a bus after every 15 minutes.

Distance covered in km	0	5	15	20	25	30	35
Time in minutes	0	15	30	45	60	75	90

From the above table we can conclude that the motion is non-uniform i.e., it covers unequal distances in equal intervals of time. The graph for a non-uniform motion will be as follows:

Distance-time graph for a non-uniform motion

• Take time along x-axis and distance along y-axis.

• Analyze the given data and make a proper choice of scale for time and distance.

• Plot the points.

• Consider any two points (A, B) on the graph.

• Draw perpendicular from A to B to x and y axes.

• Join A to C to get a right angled ACB.

Calculations:

• Write the title and scale chosen for the graph.

• Consider another two points P and Q on the graph and construct a right angled triangle PRQ.

We can infer that speed is not uniform.
Let us now see the nature of S-t graph for non-uniform motion.

Fig (a) represents the S-t graph when the speed of a moving object increases and Fig (b) represents the S-t graph when the speed of a moving object decreases.
From the nature of S-t graph we can conclude whether the object is moving with uniform speed or variable speed.

Uses of graphical representation

• Graphical representation is more informative than tables as it gives a visual representation of two quantities (e.g., speed vs. time)

At a glance a graph gives more information than a table. Both the graphs shown here represent the increasing speed.
Fig (1) gives us an idea of nature of variation of speed i.e., increase is greater in the beginning up to line t₁ and relatively lower after t₂. Similarly, fig (2) gives an idea that the increase in the speed becomes greater after t₁. Similar explanation holds good for the decreasing speed also.

• Graphs can be easily read at a glance.

• Plotting graphs takes less time and are more convenient to draw.

• By using graphs, position of any moving object at any intermediate interval can be easily determined.

• Motion of two moving objects can be easily compared.

• Graphs tell us about the nature of motion.

Speed

In an athletic meet different participants start running at the same time and cover the same distance. The person who takes the minimum time to cover the distance will be judged as the winner. Suppose Ram and Krishna take 20 minutes and 10 minutes respectively to cover a distance of 1000 m, who is faster? To find out who is faster we have to calculate the distance covered by Ram and Krishna in one minute.

i.e., Krishna covers more distance in unit time or in other words we can conclude that Krishna covered the distance with greater speed.
Speed can be defined as the distance covered by a moving object in unit time
taken. SI unit of speed is m/s or ms^-1. Speed is a scalar quantity.

 

Uniform speed

The figure shows the distance covered by a ball after every 2 seconds.

The ball covers 10 m in every 2 seconds. The speed with which the ball is moving at any point between A and E is 5 m/s. That is, the object is moving with uniform speed. An object is said to be moving with uniform speed if it covers equal distances in equal intervals of time.

Variable speed or Non-uniform speed

The previous example does hold good only if we ignore the friction or resistance offered by the surface. The distance covered varies with time.

For example, a rubber ball dropped from a height, on reaching the ground bounces off to a height less than the initial height and slowly the height decreases. The distance covered by the ball in unit time decreases. That means the speed of the ball varies from point to point. Such a speed is called a variable speed. An object is said to be moving with variable speed or non-uniform speed if it covers equal distances in unequal intervals of time or vice-versa.

Average speed and Instantaneous speed

When we travel in a vehicle the speed of the vehicle changes from time to time depending upon the conditions existing on the road. In such a situation, the speed is calculated by taking the ratio of the total distance traveled by the vehicle to the total time taken for the journey. This is called the average speed. If an object covers a distance s₁ in time t_1, distance s₂ in time t₂ and distance s_n in time t_n then the average speed is given by,
When we say that the car is traveling at an average speed of 60 km/h it does not mean that the car will be moving with the speed of 60 km/h throughout the journey. The actual speed of the car may be less than or greater than the average speed at a particular instant of time.
Hence the speed that the body possesses at a particular instant of time, is called instantaneous speed.

Distance and Displacement

Q :1 Suppose a bus starting from a terminus A travels 15000 m to reach terminus B. Then the distance covered by the bus is 15000 m. Now if the bus returns to the terminus A, then what is the distance covered by the bus during the return trip? The distance covered is 15000 m. But the total distance covered by the bus during the trip from A to B and than back to A from B is 15000 m + 15000 m = 30000 m.

A bus moving from A to B and again from B to A
Thus, the distance covered by a moving object is the actual length of the path followed by the object.
Distance is a scalar quantity. SI unit of distance is meter.
Now let us find out whether the position of the bus has changed when it is moving from the terminus A to terminus B. Yes, the position has changed, i.e., there is a displacement of 15000 m from A to B. What is the displacement of the bus during the return trip? The displacement is again 15000 m but from B to A.
Thus, displacement is the shortest distance covered by a moving object from the point of reference (initial position of the body), in a specified direction.
But the displacement when the bus moves from A → B and then from B → A is zero. SI unit of displacement in meter.
Displacement is a vector, i.e., the displacement is given by a number with proper units and direction.
To drive home the difference between displacement and distance let us consider a few more examples.

Q:2 Suppose a person moves 3 meters from A to B and 4 meters from B to C as shown in the figure. The total distance traveled by him is 7 meters. But is he actually 7 meters from his initial position? No, he is only 5 meters away from his initial position i.e., he is displaced only by 5 m, which is the shortest distance between his initial position and final position.

Now let us consider an object changing its position, with respect to a fixed point called the origin 0. x_i and x_f are the initial position and final position of the object. Then the displacement of the object = x_f - x_i.

 

Case 1

Suppose the object is moving from +1 to +4
then displacement = x_f - x_i
= +4 - (+1)
= +3

Case 2

If the object is moving from -3 to -1 then displacement = x_f - x_i
= -1 - (-3)
= 2

Case 3

If the object is moving from +4 to +2 then displacement = x_f - x_i
= +2 - (+4)
= -2.

Case 4

If the object follows the path as shown in the figure then the final position and the initial position is the same i.e., the displacement is zero.

From the above examples, we can conclude that the displacement of a body is positive if its final position lies on the right side of the initial position and negative if its final position is on the left side of its initial position. Whenever a moving object comes back to the original position then the displacement is zero.
Imagine an athlete running along a circular track of radius r in a clockwise direction starting from A.

A circular track of radius r
What is the distance covered by the athlete when he reaches the point B?
The distance covered by the athlete when he reaches the point B is equal to half of the circumference of the circular track.

Difference between distance and displacement

Distance	Displacement
It is the actual length of the path covered by a moving object	It is the shortest distance between the initial position and the final position of the moving object
It is a scalar quantity	It is a vector quantity

Motion Types

There are three types of motion:

• Translatory motion

• Rotatory motion and

• Vibratory motion

 

Translatory motion

In translatory motion the particle moves from one point in space to another. This motion may be along a straight line or along a curved path. The motion along a straight line is called rectilinear motion.
Example: A car moving on a straight road

Motion along a curved path is called curvilinear motion.
Example: A car negotiating a curve

Rotatory motion

In rotatory motion, the particles of the body describe concentric circles about the axis of motion.

Rotatory motion

Vibratory motion

In vibratory motion the particles move to and fro about a fixed point.

Simple pendulum

Motion

Introduction

     In the physical world, one of the most common phenomena is motion. The branch of Physics, which deals with the behaviour of moving objects, is known as mechanics. Mechanics is further divided into two sections namely Kinematics and Dynamics. Kinematics deals with the study of motion without taking into account the cause of motion, while Dynamics is concerned with the cause of motion, namely force. This chapter covers only the different aspects of motion without considering the cause of motion.

Motion and Rest Definition

Imagine you are travelling in a moving train. Do you observe any change in your position with respect to your co-passengers? Is there any change of scene you view through the window? The change of scene indicates that the train is moving. That is, an object is said to be in motion if it changes its position with respect to its surroundings in a given time.
We know that a table in a room is at rest i.e., its position with respect to the walls of the room does not change with time.
Those who are fond of watching night sky would have observed that the position of stars and planets change whereas we are not moving. But in reality the earth is also moving and so also all the objects on the surface of the earth. Thus, an object which appears to be at rest, may actually be in motion. Therefore, motion and rest are relative terms. Hence, to describe the motion of an object we have to specify how its position changes with respect to a fixed point called the “origin”.
We cannot tell whether an object is in motion unless we have a frame of reference. A frame of reference is another object or scene with respect to which we compare an object's position.

 

Example of Motion and Rest:

Example for frame of reference
Look at the figures. In fig.1, the car is to the right of the tree. In fig.2, after 2 seconds, the car is to the left of the tree. As the tree does not move, the car must have moved from one place to another. Therefore, here the tree is considered as the frame of reference.