Madhava Publications
1. Motion
Introduction
In our day-to-day life, we see the motion of various objects around us like the motion of cars, planes, sun, moon and people and things around us.
We all have a vague idea of Motion. But in this text, we will discuss this topic in detail with all the important aspects of the motion.
When the motion is confined to one dimension in a straight line we call this type of motion rectilinear motion. We will discuss this type of motion in this chapter.
You will have an idea of the state of motion and rest and I will introduce a very important concept of the frame of reference to discuss the state of rest or motion. Then we will learn about the kinematics of rectilinear motion in this unit. In kinematic we describe motion without going into the cause of the motion.
Motion and rest
Suppose a man is sitting in a park, he sees trees and plants around him which are at rest according to him. Now another man sitting in his car passes near that park. The man sitting in the car sees all the trees and plants moving in the opposite direction of its motion. I am sure all of you must have experienced the same situation.
Now tell me who is right and who is wrong?
Do the plants and trees are at rest or in motion?
Actually here both are right. To understand this let's first try to understand how we define something to be in motion or at rest.
The fact is that there is nothing like absolute rest or absolute motion in the whole universe. One thing which is at rest according to one observer in one frame of reference may appear to be moving according to another observer in another frame of reference.
We define something to be in rest or motion with respect to some reference which is called a frame of reference.
With respect to the man sitting in the park, trees and plants in the park appear to be at rest as they are not changing their position with respect to the man sitting there.
But since the man in the car is itself moving so when he crosses through the park, trees and plants are left behind with respect to him and as he moves ahead with some speed, he notices that trees appear to be moving with the same speed but in opposite directions.
We define all the motions with respect to some frame of reference.
We have two types of frame of reference: Inertial and Noninertial
A body is said to be at rest if its position doesn't change with respect to its surroundings, whereas when the position of a body changes with respect to its surroundings it is said to be in motion. The state of rest or motion of a body is relative to each other.
Position, path length and displacement.
Position: To describe the position of the object, we must know and be able to describe its position.
In physics, we specify a position with the help of a reference point and a set of three mutually perpendicular axes of the rectangular coordinate system.
In this chapter we are confined to one- dimension, so we need only one axis to specify the position.
In the above example Dorm is chosen as a reference point and the position of the cafeteria, physics block and library can be described with the reference of the dorm.
- The cafeteria is 1 unit left to the dorm (x=-1)
- The physics department is 2 units right to the dorm ( x=2)
- The library is 3 units right to the dorm. (x=3)
Similarly in the example below the lamp post is taken as a reference point and the position of different people is described with respect to the reference point.
Path length and displacement
Path length or distance traveled is the length of the actual path taken by the object from the initial to the final position. In the figure given below, the red represents the actual path between the start and finish points. This is the distance traveled or path length. It is a scalar quantity.
Displacement is the shortest distance between the initial and final positions. The line represented in blue is the displacement. It is a vector quantity.
The difference between Distance and Displacement is as follows:
Uniform and non-uniform motion
The motion of the object is said to be a uniform motion if the object travels an equal distance in an equal interval of time.
In the above example, a car C travels 4 m every 2 seconds throughout its journey, here car C is in uniform motion.
The motion of the object is said to be non-uniform motion if it travels an unequal distance in an equal interval of time
In the above example car C travels 1 m in the first second and then 9 m and 15 m in another consecutive second. So in this example, a car travels an unequal distance in an equal interval of time.
The distance-time graph of the uniform and non-uniform motion is given below
Speed and velocity
Speed: It is the rate of change of position of an object. It refers to how fast an object is moving. For example, if a bike covers 20 meters in 1 second. Then its speed will be 20 m/s. It is a scalar quantity.
Velocity: In contrast with speed, velocity is a vector quantity and it also tells us about the rate of change of position of the object, but it is direction aware. The magnitude of velocity is speed. In other words, we can say that velocity is the speed in a given direction.
Even if the object is in motion, its velocity can be zero. This will happen when the object returns to its initial position and hence displacement is zero, so velocity will also be zero. But speed cannot be zero if the object is in motion.
Like displacement can be negative, positive and zero, velocity can also be positive, negative and zero.
Average speed and velocity
The average speed of an object refers to the total distance it travels divided by the time which is elapsed. It is a scalar quantity.
If the object travels with speed ‘x’ from A to B and returns from B to A with speed ‘y’. Then the average speed will be
Average velocity: Average velocity is the ratio of displacement to the time taken in the entire journey. The average velocity can be zero even if the average speed is not zero.
Instantaneous velocity
The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x with respect to t.
Instantaneous velocity at any point in time is the slope of the tangent of the Distance -time graph at that point.
2. Uniform accelerated motion
Acceleration
Acceleration is the rate of change in velocity. It is a scalar quantity. Acceleration can be positive, negative and zero. It basically tells us how fast the speed of the object is changing.
In the velocity-time graph, the slope of the velocity-time graph gives acceleration.
In the figure given below, In the first case, the direction of velocity and the acceleration is opposite and hence the car is slowing down. This is an example of negative acceleration. In the second case, the direction of velocity and acceleration is in the same direction and acceleration is said to be positive acceleration and the car is speeding up.
When the motion of the object is not uniform and the speed of the object changes with time. Then motion is said to be accelerated motion.
Example of negative acceleration
In the example given above the direction of motion is toward the right and its speed decreases continuously so the acceleration is opposite
to the motion and hence this is an example of negative acceleration.
Here ,
There are two kinds of accelerated motion.
- Uniform accelerated motion
- Non-uniform accelerated motion
Uniform accelerated motion -change of velocity is constant.
If a vehicle maintains a constant or a uniform change in its velocity in a given time interval along a straight line, then the vehicle is said to have a constant acceleration motion. Its motion is called uniform accelerated motion.
In the figure is given below a car is moving with acceleration
So it changes its velocity by 2m/s every second.
Velocity-time graph and displacement time graph for uniformly accelerated motion
Freefall is an example of uniform accelerated motion
The V-t graph of uniform accelerated motion is a straight line, the slope of which gives the value of acceleration and the area under the curve gives the displacement.
Equations of uniform accelerated motion
To solve the problems related to uniform accelerated motion, we have a set of equations.
Here , v= final velocity ; u= initial velocity ; s= distance traveled
a= acceleration ; t= time taken.
Non-uniform accelerated motion:
An object is said to be in non-uniform acceleration if the velocity of the object changes by unequal amounts in equal intervals of time.
Position -Time and velocity-Time graph
We have discussed the position-time graph and velocity-time graph with each topic. Let us now see all possible types of position-time graphs with different kinds of motions.
Slope of the position-time graph gives velocity. The greater the slope faster the body moves.
Now we will recap the velocity-time graph for different kinds of motions.
Slope of the velocity-time graph gives the value of acceleration.
The area under the curve of the v-t graph gives the value of the distance traveled.
Derivation of the equation using the graphical method
We can derive the equation of accelerated motion using a graphical method by using the property of the v-t graph.
- Slope of the v-t graph gives acceleration.
- Area under the v-t graph gives distance traveled.
