Motion

"Motion is change of position of an object with time."

1.0Motion

If we look around us, we find that there are number of objects which are in motion. An object is said to be in motion if it changes its position with the passage of time.

Now observe the following bodies or objects to understand the meaning of the term "motion". Cars, cycles, motorcycles, scooters, buses, rickshaws, trucks, etc. running on the road, birds flying in the sky, fish swimming in water, all these objects are in motion. Very small objects like atoms, molecules and very large objects like planets, stars and galaxies are also in motion. Thus, all objects ranging from the smallest atom to the largest galaxy are in continuous motion. Kinematics is the science of describing the motion of objects using words, diagrams, numbers, graphs and equations. "Motion is the change in position of an object with time." Concept of a point object (or particle)

Point object

An extended object can be treated as a point object when the distance travelled by the object is much greater than its own size. A point object (or particle) is one which has no linear dimensions but possesses mass. Examples: (i) Study of motion of a train travelling from Kota to New Delhi. (ii) Revolution of earth around the sun for one complete revolution.

2.0Describing motion

When a tree is observed by an observer A standing at the railway station, the tree is at rest. This is because position of the tree is not changing with respect to the observer A (see figure).

Now, when the same tree is observed by an observer B sitting in a superfast train moving with a velocity v , then the tree is moving with respect to the observer because the position of tree is changing with respect to the observer B. Rest and motion are relative terms : There is nothing like absolute rest or motion. This means that an object can be at rest and also in motion at the same time i.e. all objects, which are stationary on earth, are said to be at rest with respect to each other, but with respect to the sun they are making revolutions. In order to study motion, therefore, we have to choose a fixed position or point with respect to which the motion has to be studied. Such a point or fixed position is called a reference point or the origin.

Discuss whether the walls of your classroom are at rest or in motion.

The walls of our classroom are at rest with respect to the ground or earth. But, they are in motion with respect to an object or an observer outside the earth. This is because the earth is moving about its own axis as well as it is revolving around the sun. Thus, the state of rest and motion are not absolute, they are relative terms.

3.0Scalar and Vector quantities

Scalar quantity

A physical quantity that is defined by its magnitude only is called a scalar quantity. Examples : Mass, time, distance, speed, work, power, energy, electric charge, volume, density, pressure, electric potential, temperature, etc.

Scalar quantities follow the algebraic (scalar) laws of addition.

Vector quantity

A physical quantity that is defined by its magnitude as well as direction is called a vector quantity. Examples : Velocity, acceleration, force, displacement, momentum, weight, torque, electric field, magnetic field, etc. Vector quantities follow the vector laws of addition. Arrows (or rays) are used to represent vectors. The direction of the arrow gives the direction of the vector. The length of the arrow is proportional to the magnitude of the vector.

4.0Difference Between Scalar And Vector Quantities

5.0Distance And Displacement

Distance

The length of the actual path between the initial and the final position of a moving object in the given time interval is known as the distance travelled by the object. Distance $=$ Length of path I (ACB) (see figure) $\sigma$ Distance is a scalar quantity. It is always taken positive. Distance is measured by odometer in vehicles. Units In SI system : metre (m). In CGS system : centimetre (cm).

Displacement

The shortest distance between the initial position and the final position of a moving object in the given interval of time is known as the displacement of the object. Displacement = Length of path II (AB) (see figure) Displacement of an object may also be defined as the change in position of the object in a particular direction. That is, Displacement of an object $=$ Final position - Initial position of the object $={\mathbf{x}}_{\mathbf{f}}-{\mathbf{x}}_{\mathbf{i}}$

• Displacement is a vector quantity. Displacement can be positive, negative or zero. Units In SI system : metre (m) In CGS system : centimetre (cm) Important points related to distance and displacement During motion, displacement of an object may be zero but the distance travelled by the object is never zero.
• Sign convention for displacement
  

Distance travelled by an object is either equal to or greater than the magnitude of displacement of the object. (G) Distance is equal to magnitude of displacement when a body moves in a straight line in a particular direction or it is in uniform motion.

Q. A honeybee leaves the hive and travels 2 m as it returns to the hive. Is the displacement for the trip the same as the distance travelled? If not, why not?

Hive

Numerical Ability

• Motion of a particle is shown below on a number line. Find the displacement from (a) A to B (b) B to C (c) overall journey. Also, find distance for overall journey.
  
  Decode the problem: To find the displacement, we should know the initial and the final position of the particle. Apply the formula - Displacement $={\mathbf{x}}_{\mathbf{f}}-{\mathbf{x}}_{\mathbf{i}}$ Distance $=$ Length of actual path travelled Solution: (a) At point A, initial position of the particle ( ${\mathrm{x}}_{\mathrm{i}}$ ) is 0 m and at point $B$, the final position of the particle $(xf)$ is +6 m . Displacement from $A$ to $B=xf-x_i=(6)-(0)=+6m$ (b) At point B, initial position of the particle ( ${\mathrm{x}}_{\mathrm{i}}$ ) is 6 m and at point $C$, the final position of the particle $\left({\mathrm{x}}_{\mathrm{f}}\right)$ is -6 m . Displacement from B to $C=x_f-x_i=(-6)-(6)=-12\mathbf{m}$ (c) At point A, initial position of the particle ( ${\mathrm{xi}}_{\mathrm{i}}$ ) is 0 m and at point C , the final position of the particle $\left({\mathrm{x}}_{\mathrm{f}}\right)$ is -6 m . Displacement of overall journey (i.e. A to B, B to C) $=x_f-x_i=(-6)-(0)=-6\mathbf{m}$ Here, distance can also be found by adding positive values of displacement AB & displacement BC . i.e., Distance travelled during overall journey $=AB+BC$ $=6+12=18\mathrm{m}$ Here Distance > |Displacement|
• A body falls from a height of 3m. Find displacement and distance.
  
  Solution- The initial position of the particle ( ${\mathrm{xi}}_{\mathrm{i}}$ ) is 3 m (the body is 3 m above the ground) and the final position of the particle ( xf ) is 0 m . (see figure) Displacement $=x_f-x_i=0-3=-3m$, Distance $=3\mathrm{m}$ (see figure)
• A particle moves along a circular path as shown in figure. Find distance travelled and displacement.
  
  Decode the problem: To find the displacement, we should know the initial and the final position of the particle. Apply the formula - Displacement $={\mathbf{x}}_{\mathbf{f}}-{\mathbf{x}}_{\mathbf{i}}$ Distance $=$ Length of actual path travelled Solution- The shortest distance is the straight line between points $A$ and $B$. Displacement $=$ diameter $\mathrm{AB}=2R$ To find the distance, we should know the formula of the circumference of the circle. Circumference of the circle $=2\pi r$ Distance travelled $=\frac{1}{2}\times$ (circumference of the circle $)$ $=\frac{1}{2}(2\pi \mathrm{R})=\pi \mathrm{R}$

Remember

• Always take care of the sign convention while finding the displacement of the body.
• Always take care of the direction (north, south, east, west) given in the problem.
  
• For a Circle : Diameter $=2\mathrm{r}$ Circumference $=2\pi$ r or $\pi \mathrm{d}(\pi =\frac{22}{7}$ or 3.14$)$
  
• Quadrilateral Perimeter of quadrilateral $=\mathrm{AB}+\mathrm{BC}+\mathrm{CD}+\mathrm{DA}$ Perimeter of rectangle $=2(l+b)$ Perimeter of square $=4\times$ side
  
• Pythagoras theorem $(\text{ Hypotenuse }{)}^2=(\text{ Perpendicular }{)}^2+(\text{ Base }{)}^2$ ${\mathrm{AC}}^2={\mathrm{BC}}^2+{\mathrm{AB}}^2$
  
• Whenever a particle changes its direction, distance is greater than displacement.
• A body moving in a circular path when reaches its original position after one round, then the displacement at the end of one round is zero, but the distance travelled by it is equal to the circumference of circular path.
• A body moving in a circular path when covers $\frac{1}{4}$ th of the circle then, the distance travelled by it is $\frac{\pi r}{2}$ and displacement is $\sqrt{2\mathrm{r}}$.
• A body moving in a circular path when covers $\frac{3}{4}$ th of the circle then, the distance travelled by it is $\frac{3\pi \mathrm{r}}{2}$ and displacement is $\sqrt{2\mathrm{r}}$.

Comparison Between Distance And Displacement

6.0Uniform and Non-Uniform motion

A moving body may cover equal distances in equal intervals of time or different distances in equal intervals of time. On the basis of above assumption, the motion of a body can be classified as uniform motion and non-uniform motion.

1. Uniform Motion

When a body covers equal distances in equal intervals of time, however small may be the time intervals, in a particular direction, the body is said to describe a uniform motion. (see figure).