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1. Motion
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. Describing 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). Describing motion 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. Explanation 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.
2. Scalar and vector quantities
Scalar quantity A physical quantity that is defined by its magnitude only is called a scalar quantity.
I Concept of distance and displacement 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 = xf – xi 👉 Displacement is a vector quantity. (Vertical direction) x-axis
- Sign convention for displacement (Horizontal direction) y-axis 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. Distance travelled by an object is either equal to or greater than the magnitude of displacement of the object. ☞ 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.
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 Building concepts 2 Explanation No, the displacement and the distance are not same. This is because the displacement is the change of position of object in motion while distance is length of path travelled by it. Here, the distance travelled = 2 m While the displacement = 0, because the position of honey bee is not changed.
1. 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. Intermediate position
0 2 4 6 8 Final position Initial position B
A B R R Numerical Ability 1 (3) Solution The shortest distance is the straight line between points A and B. Displacement = diameter AB = 2R To find the distance, we should know the formula of the circumference of the circle. Circumference of the circle = 2πr Distance travelled = × (circumference of the circle) = (2πR) = πR ★ Always take care of the sign convention while finding the displacement of the body. N S
W E ★ Always take care of the direction (north, south, east, west) given in the problem. ★ For a Circle : Diameter = 2r r Circumference = 2πr or πd ★ Quadrilateral A B C D Perimeter of quadrilateral = AB + BC + CD + DA Perimeter of rectangle = 2( l + b) Perimeter of square = 4 × side B C A h b p ★ Pythagoras theorem (Hypotenuse)^2 = (Perpendicular)^2 + (Base)^2 AC^2 = BC^2 + AB^2
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. 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). Time (in second) 0 1 2 3 4 5 6 Distance covered (in metre) 0 1 0
10 20 30 40 50 60 Time (s) Distance (m)
Distance-time graph for uniform motion 👉 Uniform motion always takes place in a straight line. In uniform motion, velocity of particle remains constant i.e., its magnitude as well as direction are constant. In uniform motion, average speed/velocity is equal to instantaneous speed/velocity at any point of time. Examples of uniform motion (i) An aeroplane flying at a speed of 600 km/h along north. (ii) A train running at a speed of 120 km/h along east. (iii) Light energy travelling at a speed of 3 × 10^8 m/s in vacuum. Non-uniform motion When a body covers unequal distances in equal intervals of time, the body is said to be moving with a non-uniform motion. (see figure) 👉 Any motion along a curved path is always non-uniform motion. Also, any motion in which particle changes its direction is also non-uniform motion. Time (in second) 0 1 2 3 4 Distance (in metre) 0 1 4 9 1 6 0 1 2 3 4 5 6 1 4 9
Speed in m/s = × speed in km/h ; Km h–
1. A car travels first half distance with a uniform speed u and next half distance with a uniform speed v. Find its average speed. Solution A d d/ d/ t 2 t 1 u v B Numerical Ability 2 (1) Total distance = + = d [See figure] Total time = t 1 + t 2 = t ∴t 1 = ...(i) t 2 = ...(ii)
Vav = = Putting the value of equation (i) and (ii), Vav (Taking d/2 common out of bracket in the denominator)
2. A car travels first half time with a uniform speed u and next half time with a uniform speed v. Find its average speed. Solution Since the car travels with a constant speed therefore, to find the distance the formula to be used is Distance = speed × time. , [See figure] t d 2 d 1 u v A t/ t/ B Numerical ability 2 (2) Total distance d = d = Total time = t
2 km/min = = = 33.3 m/s Thus, the average speeds of the bicycle, the athlete and the car are 5 m/s, 7 m/s and 33.3 m/s respectively. So the car is the fastest, and the bicycle is the slowest.
2. On a 120 km track, a train travels the first 30 km with a uniform speed of 30 km/h. How fast must the train travel the next 90 km so as to average 60 km/h for the entire trip? Solution Given; Total distance d = 120 km, Average speed Vav = 60 km/h, Total time = t =? Vav = or t = Putting the values, t = = 2 h ...(i) Distance travelled in first part of trip, d 1 = 30 km, Speed in first part of the trip, v 1 = 30 km/h Time taken in first part of trip, t 1 = ?, t 1 = Putting the values, t 1 = = 1 h Time taken to complete second part of the trip, t 2 = t – t 1 = 2 – 1 = 1 h Distance to be covered in second part of the trip, d 2 = 90 km, Required speed in second part, v 2 =? Speed = , ∴ v 2 = = = 90 km/h Check the unit Use km/hr as the distance is given in km and time is given in hrs. 3. A bus going from Kota to Jaipur passed the 100 km, 160 km and 220 km points at 10:30 am, 11:30 am and 1:30 pm. Find the average speed of the bus during each of
the following intervals: (a) 10.30 am to 11.30 am, (b) 11.30 am to 1.30 pm and (c) 10.30 am to 1.30 pm. Solution (a) The distance covered between 10:30 am and 11: am is 160 km – 100 km = 60 km. The time interval is 1 hour. The average speed during this interval is – v 1 = = 60 km/h (b) The distance covered between 11:30 am and 1:30 pm is 220 km – 160 km. = 60 km. The time interval is 2 hours. The average speed during this interval is – v 2 = = 30 km/h (c) The distance covered between 10:30 am and 1:30 pm is 220 km – 100 km = 120 km. The time interval is 3 hours. The average speed during this interval is – v 3 = = 40 km/h
6. Velocity
The velocity of a body is the displacement of a body per unit time. The displacement covered by a body per unit time or the speed of a body in specified direction is called velocity. 👉 Velocity is a vector quantity. It can be positive, negative or zero (see figure). y-axis (Vertical direction) x-axis
1s A D C B 5m 3m 1s 7m 1s motion Body moving with non-uniform velocity When a body covers equal distances in equal intervals of time, but its direction changes, then the body is said to be moving with variable velocity. Example : In circular motion, a particle may have constant speed but its direction changes continuously thus, its velocity is non-uniform (see figure). A D C B 1s 1s 1s
1s 5m 5m 5m 5m Body moving with variable velocity Conditions for variable velocity (i) It should cover unequal displacements in equal intervals of time. (ii) It should cover equal distances in equal intervals of time but its direction must change. Examples (i) A car running towards north on a busy road has a variable velocity as the displacement covered by it per unit time changes with change in the road condition. (ii) The blades of a rotating ceiling fan, a person running around a circular track with constant speed etc. are examples of variable velocity. Average velocity Total displacement of a particle divided by total time taken is called average velocity. Vav = 👉 Average speed is always greater than or equal to magnitude of average velocity. Average speed is equal to average velocity when particle moves in a straight line without change in direction. Instantaneous velocity The velocity of a body at any particular instant of time during its motion is called the instantaneous velocity of the body. A particle moves along a path ABC as shown in figure. The time taken during the journey is 2 seconds. Find the average speed and average velocity during the journey.
