Motion in Two and Three Dimensions Chapter 3 Motion in Two and Three Dimensions

Types of physical quantities In physics, quantities can be divided into such general categories as scalars, vectors, matrices, etc. Scalars – physical quantities that can be described by their value (magnitude) only Vectors – physical quantities that can be described by their value and direction

Vectors Vectors are labeled either a or Vector magnitude is labeled either |a| or a Two (or more) vectors having the same magnitude and direction are identical

Vector sum (resultant vector) Not the same as algebraic sum Triangle method of finding the resultant: Draw the vectors “head-to-tail” The resultant is drawn from the tail of A to the head of B R = A + B B A

Addition of more than two vectors When you have many vectors, just keep repeating the process until all are included The resultant is still drawn from the tail of the first vector to the head of the last vector

Commutative law of vector addition A + B = B + A

Associative law of vector addition (A + B) + C = A + (B + C)

Negative vectors Vector (- b) has the same magnitude as b but opposite direction

Vector subtraction Special case of vector addition: A - B = A + (- B)

Multiplying a vector by a scalar The result of the multiplication is a vector c A = B Vector magnitude of the product is multiplied by the scalar |c| |A| = |B| If the scalar is positive (negative), the direction of the result is the same as (opposite to that) of the original vector

Vector components Component of a vector is the projection of the vector on an axis To find the projection – drop perpendicular lines to the axis from both ends of the vector – resolving the vector

Vector components

Unit vectors Unit vector: Has a magnitude of 1 (unity) Lacks both dimension and unit Specifies a direction Unit vectors in a right-handed coordinate system

Adding vectors by components In 2D case:

Chapter 3 Problem 42 Vector A has magnitude 1.0 m and points 35° clockwise from the x-axis. Vector B has magnitude 1.8 m. Find the direction of B such that A + B is in the y-direction.

Position The position of an object is described by its position vector,

Displacement The displacement vector is defined as the change in its position,

Velocity Average velocity Instantaneous velocity

Instantaneous velocity Vector of instantaneous velocity is always tangential to the object’s path at the object’s position

Acceleration Average acceleration Instantaneous acceleration

Acceleration Acceleration – the rate of change of velocity (vector) The magnitude of the velocity (the speed) can change – tangential acceleration The direction of the velocity can change – radial acceleration Both the magnitude and the direction can change

Chapter 3 Problem 23 What are (a) the average velocity and (b) the average acceleration of the tip of the 2.4-cm-long hour hand of a clock in the interval from noon to 6 PM? Use unit vector notation, with the x-axis pointing toward 3 and the y-axis toward noon.

Projectile motion A special case of 2D motion An object moves in the presence of Earth’s gravity We neglect the air friction and the rotation of the Earth As a result, the object moves in a vertical plane and follows a parabolic path The x and y directions of motion are treated independently

Projectile motion – X direction A uniform motion: ax = 0 Initial velocity is Displacement in the x direction is described as

Projectile motion – Y direction Motion with a constant acceleration: ay = – g Initial velocity is Therefore Displacement in the y direction is described as

Projectile motion: putting X and Y together

Projectile motion: trajectory and range

Projectile motion: trajectory and range

Chapter 3 Problem 33 A carpenter tosses a shingle horizontally off an 8.8-m-high roof at 11 m/s. (a) How long does it take the shingle to reach the ground? (b) How far does it move horizontally?

Uniform circular motion A special case of 2D motion An object moves around a circle at a constant speed Period – time to make one full revolution The x and y directions of motion are treated independently

Uniform circular motion Velocity vector is tangential to the path From the diagram Using We obtain

Centripetal acceleration

Centripetal acceleration During a uniform circular motion: the speed is constant the velocity is changing due to centripetal (“center seeking”) acceleration centripetal acceleration is constant in magnitude (v2/r), is normal to the velocity vector, and points radially inward

Chapter 3 Problem 38 How fast would a car have to round a 75-m-radius turn for its acceleration to be numerically equal to that of gravity?

Relative motion Reference frame: physical object and a coordinate system attached to it Reference frames can move relative to each other We can measure displacements, velocities, accelerations, etc. separately in different reference frames

Relative motion If reference frames A and B move relative to each other with a constant velocity Then Acceleration measured in both reference frames will be the same

Questions?

Answers to the even-numbered problems Chapter 3 Problem 28: 196 km/h

Answers to the even-numbered problems Chapter 3 Problem 30: 3.6 ˆi − 4.8 ˆj m/s2 6.0 m/s2; 53°

Answers to the even-numbered problems Chapter 3 Problem 34: 1.5 m