# MOTION IN A PLANE

Syllabus Summary: Process of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications.Scalar and vector quantities; position and displacement vectors, general vectors and their notations; equality of vectors, multiplication of vectors by a real number; addition and subtraction of vectors, relative velocity, Unit vector; resolution of a vector in a plane, rectangular components, Scalar and Vector product of vectors.
Motion in a plane, cases of uniform velocity and uniform acceleration-projectile motion, uniform circular motion.

## Lecture - 1

00:01:42 Multiplication of a vector by a scalar

00:09:00 Unit Vector & Finding a new vector in direction of given vector

00:16:35 General Vector v/s Unit Vector, Direction Ratios, Direction Cosines

00:25:50 Meaning of Direction Cosines

00:31:16 Types of Vectors

00:31:20 Equal Vectors

00:34:45 Negative Vectors

00:36:45 Collinear Vectors and Parallel Vectors

00:42:30 Coinitial Vector

## Lecture - 2

00:00:02 Every two dimensional vector is a sum of individual one dimensional vectors

00:03:16 Meaning of resultant vector

00:04:22 Vector Addition

00:08:22 Parallelogram Law of Vector Addition

00:15:30 Triangle Law of Vector Addition

00:20:56 Direction of a vector in terms of angle made with x-axis and y-axis

00:26:46 Example 4.1 Rain is falling vertically with a speed of 35 m/s . Winds starts blowing in after sometime with a speed of 12 m/s east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella ?

00:39:36 Example 4.2 Find the magnitude and direction of the resultant of two vectors A and B in terms of their magnitudes and angle θ between them.

00:53:06 Example 4.3 A motorboat is racing towards north at 25 km/h and the water current in that region is 10 km/h in the direction of 60° east of south. Find the resultant velocity of the boat.

01:01:15 Use of log tables (logarithms) while solving Physics numericals/questions

## Lecture - 3

• 00:02:11 Displacement in two dimension (Displacement in a plane)

00:09:02 Average Velocity & Instantaneous Velocity in two dimension

00:17:55 Average Acceleration & Instantaneous Acceleration in two dimension

00:24:25 Equations of motion in two dimensions

00:38:19 Example 4.4 The position of a particle is given by r=3.0ti+2.0t^2j+5.0k, where t is in seconds and the coefficients have the proper units for r to be in metres. (a) Find v(t) and a(t) of the particle. (b) Find the magnitude and direction of v(t) at t = 1.0 s.

00:48:49 Working with Trigonometry Table of all angles

00:52:39 Example 4.5 A particle starts from origin at t = 0 with a velocity 5.0 î m/s and moves in x-y plane under action of a force which produces a constant acceleration of (3.0i+2.0j) m/s^2. (a) What is the y-coordinate of the particle at the instant its x-coordinate is 84 m ? (b) What is the speed of the particle at this time ?

01:02:49 Using Log Table in Calculations

## Lecture - 4

00:00:10 Projectile Motion – Two dimensional motion under gravity

00:05:02 Animations for Projectile motion

00:26:02 Derivations for Projectile Motion

00:35:29 Relation to find time of Maximum Height

00:38:59 Relation to find maximum height

00:43:59 Equation of path traced by object in projectile motion (Equation of parabola)

00:40:00 Total Time taken

00:40:01 Horizontal Range of Projectile

01:02:59 Example 4.9 A cricket ball is thrown at a speed of 28 m/s in a direction 30° above the horizontal. Calculate (a) the maximum height, (b) the time taken by the ball to return to the same level, and (c) the distance from the thrower to the point where the ball returns to the same level.

## Lecture - 5

• 00:07:00 Horizontal Projectile Motion Introduction with animations

00:10:30 Example 4.8 (Explanations for Horizontal Projectile Motion) A hiker stands on the edge of a cliff 490 m above the ground and throws a stone horizontally with an initial speed of 15 m/s. Neglecting air resistance, find the time taken by the stone to reach the ground, and the speed with which it hits the ground. (Take g = 9.8 m/s^2)

00:25:40 Example 4.7 (Maximum Range of a Projectile with animations) Galileo, in his book Two new sciences, stated that “for elevations which exceed or fall short of 45° by equal amounts, the ranges are equal”. Prove this statement.

## Lecture - 6

• 00:00:20 Algebraical and Graphical Meaning of Vector Subtraction

00:04:40 Vector Subtraction using Triangle Law of Vector Addition

00:08:10 Vector Subtraction using Parallelogram Law of Vector Addition

00:18:40 Introduction to Uniform Circular Motion

00:19:52 Animation for Uniform Circular Motion

00:25:20 Real Life Application of Uniform Circular Motion

00:29:40 Derivations for Uniform Circular Motion

00:50:29 Direction of Average Velocity, Average Acceleration and Centripetal Acceleration

## Lecture - 7

00:00:20 Animation for Centripetal Acceleration Under Uniform Circular Motion

00:20:20 Angular Displacement

00:25:22 Angular Speed (Angular Velocity)

00:27:52 Relation between Angular Speed and Linear Speed

00:29:52 Relation between Centripetal Acceleration and Angular Speed

00:31:12 Time Period and Frequency in Uniform Circular Motion

00:33:42 Linear Speed in terms of Frequency

00:34:22 Angular Speed in terms of Frequency

00:34:52 Centripetal Acceleration in terms of Frequency

## Lecture - 8

• 00:00:25 Example 4.6 Example 4.6 Rain is falling vertically with a speed of 35 m/s . A woman rides a bicycle with a speed of 12 m/s in east to west direction. What is the direction in which she should hold her umbrella ?

00:13:45 Example 4.10 Example 4.10 An insect trapped in a circular groove of radius 12 cm moves along the groove steadily and completes 7 revolutions in 100 s. (a) What is the angular speed, and the linear speed of the motion? (b) Is the acceleration vector a constant vector ? What is its magnitude ?

00:18:55 Exercise 4.1 State, for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.

00:20:05 Exercise 4.2 Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.

00:20:55 Exercise 4.3 Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.

00:21:55 Exercise 4.4 State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars, (b) adding a scalar to a vector of the same dimensions , (c) multiplying any vector by any scalar, (d) multiplying any two scalars, (e) adding any two vectors, (f) adding a component of a vector to the same vector.

00:27:35 Exercise 4.5 Read each statement below carefully and state with reasons, if it is true or false : (a) The magnitude of a vector is always a scalar, (b) each component of a vector is always a scalar, (c) the total path length is always equal to the magnitude of the displacement vector of a particle. (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time, (e) Three vectors not lying in a plane can never add up to give a null vector.

00:37:55 Exercise 4.7 Given a + b + c + d = 0, which of the following statements are correct : (a) a, b, c, and d must each be a null vector, (b) The magnitude of (a + c) equals the magnitude of ( b + d), (c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d, (d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear ?

00:48:37 Exercise 4.8 Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 4.20. What is the magnitude of the displacement vector for each ? For which girl is this equal to the actual length of path skate ?

00:51:18 Exercise 4.9 A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (a) net displacement, (b) average velocity, and (c) average speed of the cyclist ?

00:55:20 Exercise 4.6 Establish the following vector inequalities geometrically or otherwise : (a) |a+b| less than |a| + |b| (b) |a+b| greater than or equal ||a| −|b|| (c) |a−b| less than |a| + |b| (d) |a−b| Greater than or equal ||a| − |b||

## Lecture - 9

• 00:00:15 Exercise 4.10 On an open ground, a motorist follows a track that turns to his left by an angle of 600 after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.

00:23:13 Exercise 4.11 A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi, (b) the magnitude of average velocity ? Are the two equal ?

00:27:50 Exercise 4.12 Rain is falling vertically with a speed of 30 m per sec. A woman rides a bicycle with a speed of 10 m per sec in the north to south direction. What is the direction in which she should hold her umbrella ?

00:39:45 Exercise 4.13 A man can swim with a speed of 4.0 km/h in still water. How long does he take to cross a river 1.0 km wide if the river flows steadily at 3.0 km/h and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank ?

## Lecture - 10

00:00:15 Exercise 4.14 In a harbour, wind is blowing at the speed of 72 km/h and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of 51 km/h to the north, what is the direction of the flag on the mast of the boat ?

00:22:26 Exercise 4.15 The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of 40 m/s can go without hitting the ceiling of the hall ?

00:34:01 Exercise 4.16 A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball ?

00:40:54 Exercise 4.17 A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone ?

00:45:20 Exercise 4.18 An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity.

00:49:56 Exercise 4.19 Read each statement below carefully and state, with reasons, if it is true or false: (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector

00:53:32 Exercise 4.20 The position of a particle is given by r=3.0i-2.0t^2j+4.0k m where t is in seconds and the  coefficients have the proper units for r to be in metres. (a) Find the v and a of the particle? (b) What is the magnitude and direction of velocity of the particle at t = 2.0 s ?

01:01:54 Exercise 4.21 A particle starts from the origin at t = 0 s with a velocity of 10.0 j m/s and moves in the x-y plane with a constant acceleration of (8.0i+2.0)j m/s^2 (a) At what time is the x- coordinate of the particle 16 m? What is the y-coordinate of the particle at that time? (b) What is the speed of the particle at the time ?