Describing Motion: Motion along a Straight Line, Distance-Time

Describing Motion: Look in the room around you. Some objects may be at rest while some may be moving. A body is said to be moving if it changes its position with time, while a body is considered to be at rest if its position remains the same with time. All the planets in outer space, stars, galaxies, atoms, and molecules are continuously moving. Flying birds, swimming fishes, walking dogs, and running horses are all in motion. But the state of rest or the state of motion is all dependent on the observer’s perspective.

For us, day and night occur due to the sun’s motion, but is it truly so? The changes in season or the sunrise and sunset are due to the earth’s motion, but why do we not feel the earth moving? Why do trees appear to be running back when we look at them from trains running rapidly in the forward direction? Why is it that all the people in a moving car appear to be moving to a person watching it from the roadside? But to a person inside the car, all his fellow passengers appear to be at rest! Let get the answers to our questions and learn in detail about the state of motion.

What is Motion?

Motion is associated with the word “movement.” It is described as the change in the position or orientation of an object with respect to its surroundings in a short interval of time. Motion in itself is a quite complex phenomenon, and it can be of the following three types:

1. Linear: In this motion, particles move from one point to another along a straight line, e.g., the motion of a car along a straight road.
2. Circular: In this motion, particles move from one point to another along a circular path, e.g., the motion of a giant wheel ride in an amusement park.
3. Rotational: In this motion, particles rotate about their axis,e.g., the earth’s motion about its axis.

In general, we deal with the motion of objects along a straight line. To actually study the motion of an object, we will first need to choose a starting point. The starting point, also called the reference point, can be chosen as per our convenience, and it serves as an origin that helps describe the motion of an object.

Learn Newton’s Laws of Motion here

Motion along the Straight Line

Motion along a straight line path is probably the simplest motion to understand. Some terms that are intricate for the description of straight-line motion are:
Distance is the length of the total path covered by a moving object along a straight line. Distance is a scalar quantity meaning that it has only a numerical value but no direction. The magnitude of distance can never be zero.

Displacement is the length of the shortest path between the initial and final positions of a moving object. Displacement is a vector quantity meaning that it has both numerical value and direction. The displacement of an object can be zero.
In general, the displacement of an object is always less than its distance, although for an object moving along a straight line, the distance covered by it is equal to its displacement.
Uniform Motion: If a moving object covers equal distance in equal intervals of time, its motion is said to be uniform.
Non-uniform Motion: If a moving object covers an unequal distance in equal intervals of time or equal distance in unequal intervals of time or unequal distance in unequal intervals of time, the object is said to performing the non-uniform motion.

Average Speed

While discussing motion, often we need to get an idea about the rate at which different objects are moving. Some objects might be slower than others, and some might be faster. The rate at which different objects are moving can be different or the same.
The speed of an object is the quantity that we use to compare the rate at which objects are moving. Speed is defined as the distance covered by an object per unit of time. The SI unit of speed is metre per second \((\rm{m/s})\). Speed is a scalar quantity, i.e. it only has magnitude.
\({\text{Speed = }}\frac{{{\text{Distance}}}}{{{\text{Time}}}}\)
Most of the objects perform the non-uniform motion, meaning that their speed varies throughout the motion. In such a situation, we calculate the average speed to compare the motion of different objects.
The average speed is equal to the total distance covered by an object divided by the total time for which the object was in motion. Mathematically,
\({\text{Average}}\,{\text{speed = }}\frac{{{\text{Total}}\,{\text{distance}}}}{{{\text{Total}}\,{\text{time}}}}\)
The SI unit of average speed is the same as the unit of speed, i.e. metre per second.

Average Velocity

The speed of an object does not give us an idea about the direction in which an object is moving. To get a comprehensive idea about the rate of motion of an object throughout its journey, we need a quantity that will account for the direction of the object. Velocity is such a quantity. It is defined as the rate of change of displacement per unit time. It is the speed of an object in a specific direction. It is a vector quantity meaning it has both magnitude and direction. The SI unit of velocity is a metre per second. Since velocity can be varied by varying either the speed or direction or both of a moving object, we use the quantity average velocity to compare the motion of different objects.
Average velocity is defined as the arithmetic mean of an object’s initial and final velocity for a given time period. Mathematically,
\({\text{Average}}\,{\text{velocity = }}\frac{{{\text{initial}}\,{\text{speed + final}}\,{\text{speed}}}}{{\text{2}}}\)
The SI unit of average velocity is the same as the unit of velocity, i.e. metre per second.

Acceleration

For objects undergoing non-uniform motion, velocity does not remain constant. The quantity that determines the rate of change in the velocity of an object is the acceleration. It is defined as the change in velocity per unit of time. Mathematically,
\({\text{acceletation = }}\frac{{{\text{Final}}\,{\text{velocity – initial}}\,{\text{velocity}}}}{{{\text{time}}}}\)
The SI unit of acceleration is \(\rm{m/s}^2\). Acceleration is a vector quantity that has both magnitude and direction.
Acceleration of an object is positive in the direction of velocity and negative in the opposite direction of velocity.

Graphical Representation

Studying the motion of an object throughout the entire journey becomes easier if we plot a graph. Graphs provide a better understanding and a much clear picture of how an object is moving. For example, graphs in a cricket match are often plotted, run rate per over, to see how a team performed.
For any moving object in general, “displacement vs time” and “velocity vs time” -line graphs are plotted. Such graphs are often seen in car racing events where the motion of the cars are studied per lap.

Distance-Time Graph

Using an appropriate scale, we can plot the change in position of an object with time. In such a graph, distance is plotted along the \(y-\)axis, and time is plotted along the \(x-\)axis. We can plot such graphs to study the conditions of an object in uniform or non-uniform motion.

The above graph is plotted for an object undergoing uniform motion, i.e. object is covering equal distances in equal time intervals. As we can see from the graph, the distance travelled by the object is directly proportional to the time taken. For an object moving at a uniform speed, the distance-time graph is a straight line passing through the middle of the \(x-\)and \(y-\)axis, i.e. making an angle of \(45^\circ\) from both axis.
From the graph, we can see that the distance covered by the object is increasing uniformly. Here we can use the word uniform velocity in place of uniform speed if the magnitude of displacement of the object is kept equal to distance along the \(y-\)axis.

Velocity-Time Graph

Acceleration is the rate of change of velocity of an object. Using a velocity-time graph, we can calculate the acceleration of an object. In such a graph, velocity is plotted along the \(y-\)axis and time is plotted along the \(x-\)axis. Such graphs can be plotted for uniformly and non-uniformly accelerated motion of an object.
For a particle in linear motion along one direction, the velocity-time or speed-time graphs for different situations can be given as:

Calculation of Speed

Speed is the rate of change of distance with time. Using a distance-time graph, we can calculate the speed of an object. For this, take a distance-time graph as shown below:

Take a small region “\(AB\)” of the distance-time graph, from point \(A\), draw a line parallel to the \(x-\)axis and from point \(B\), draw a line parallel to the \(y-\)axis. The two lines meet at a point \(C\), and we get a triangle \(ABC\).
The graph shows that the object moves from point \(A\) to point \(B\); it covers the distance \(BC\) in the time interval \(AC\).
Here, \(AC\) represents the time interval \(t_2 – t_1\) and \(BC\) represents the distance covered, i.e., \(S_2 – S_1\). We know that the speed of an object is the ratio of the distance covered over the time taken. Thus,
\({\text{Speed = }}\frac{{{\text{Distance}}}}{{{\text{time}}}} = \frac{{{S_2} – {S_1}}}{{{t_2} – {t_1}}}\)
Thus, the speed of an object is equal to the slope of the displacement time curve.
Similarly, a graph can be plotted to represent the accelerated motion of an object. In this case, the shape of the curve will vary in each time interval, and one such graph has been shown below:

From the graph, we can conclude that a non-linear variation of distance covered by the object with time exists. Thus, the object’s motion is non-uniform.

Calculation of Acceleration and Distance

For uniformly accelerated motion, consider the graph shown below:

The velocity-time graph for the object is a straight line passing right through the centre of the \(x\) and \(y-\)axis, meaning that the object’s velocity changes equally in equal intervals of time. From the graph, we can conclude that the object’s change in velocity is directly proportional to the time. To calculate the object’s acceleration, consider two points \(A\) and \(E\) on the curve and draw perpendiculars from these two points on the \(x-\)and \(y-\)axis. The interval \(FG\) represents the change in velocity \(v_2 – v_1\) and the interval \(BC\) represents the time interval \(t_2 – t_1\).
Thus, the acceleration of the object can be given as:
\({\text{acceleration = }}\frac{{{\text{change}}\,{\text{in}}\,{\text{velocity}}}}{{{\text{time}}\,{\text{interval}}}} = \frac{{{v_2} – {v_1}}}{{\;{t_2} – {t_1}}}\)
Thus, the acceleration of an object is equal to the slope of the velocity-time curve.
We can also calculate the displacement covered by an object from the velocity-time graph. We know that the displacement of an object is equal to the product of time and uniform velocity. Thus, the magnitude of displacement can be calculated by measuring the area within the velocity-time curve and the time axis. We can calculate the total distance covered or the magnitude of the displacement of an object. To do so, take the graph given above.
\({\text{Distance}}\,{\text{covered = Area}}\,{\text{under}}\,{\text{velocity – time}}\,{\text{curve}}\)
\({\text{ = Area}}\,{\text{of}}\,{\text{trapezium}}\,ABCE\)
\( = \frac{1}{2} \times \left( {AB + EC} \right) \times BC\)
\( = \frac{1}{2} \times \left( {{v_1} + {v_2}} \right) \times \left( {{t_2} – {t_1}} \right)\)
The velocity-time graph for an un-accelerated object or  an object moving with zero acceleration can be given as:

For an object moving with constant velocity, the velocity-time curve is a straight line parallel to \(x-\)axis where the curve’s height will remain the same.
The velocity-time graph for an object in non-uniform acceleration can be given as:

Summary

Motion is described as the change in the position or orientation of an object with respect to its surroundings in a short interval of time.
The average speed equals the total distance covered by an object divided by the total time for which the object was in motion. Average velocity is the arithmetic mean of an object’s initial and final velocity for a given time. Acceleration is defined as the change in velocity per unit of time. Studying the motion of an object throughout the entire journey becomes easier if we plot a graph. Graphs provide a better understanding and a much clear picture of how an object is moving.

1. Speed is the rate of change of distance with time.  Using a distance-time graph, we can calculate the speed of an object.
   \({\text{Speed = }}\frac{{{\text{Distance}}}}{{{\text{time}}}} = \frac{{{S_2} – {S_1}}}{{{t_2} – {t_1}}}\)
   Thus, the speed of an object is equal to the slope of the displacement time curve.
2. The acceleration of the object can be given as:
   \({\text{acceleration = }}\frac{{{\text{change}}\,{\text{in}}\,{\text{velocity}}}}{{{\text{time}}\,{\text{interval}}}} = \frac{{{v_2} – {v_1}}}{{\;{t_2} – {t_1}}}\)
   Thus, the acceleration of an object is equal to the slope of the velocity-time curve.
3. The total distance(or magnitude of displacement) can be calculated by measuring the area within the velocity-time curve and the time axis.
   \({\text{Distance}}\,{\text{covered = Area}}\,{\text{under}}\,{\text{velocity – time}}\,{\text{curve}}\)

Frequently Asked Questions on Describing Motion

Q.1. Define average velocity.
Ans: The average velocity is equal to the mean of its initial and final velocity within the given time interval.

Q.2. How can you calculate the speed of an object using a distance-time graph?
Ans: The slope of the distance-time graph will give you the speed of the object.

Q.3. Draw the velocity-time graph of an object whose velocity is increasing.
Ans: The graph for an object whose velocity is increasing can be given as:

Q.4. Define displacement.
Ans: Displacement of an object is the shortest distance between its initial and final position.

Q.5. Can you calculate the distance covered by an object using a velocity-time graph?
Ans: Yes, the distance an object covers equals the area between the velocity-time curve and the time axis.

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