Chapter 3: Vectors & Two-Dimensional Motion Vectors and Their Properties All the physical quantities we will learn are classified either as a vector or scalar quantity. A scalar is specified by its magnitude, while a vector by its magnitude AND its direction. Examples Scalar : temperature, speed, mass, volume, length, etc. Vector : displacement, velocity, acceleration, force, etc.

Vectors and Their Properties Notation Equality of two vectors if their magnitudes and the directions are the same. Addition of vectors (geometrical method) Commutative law of addition:

Vectors and Their Properties Negative of a vector is defined as the vector that gives zero when added to has the same magnitude but opposite direction of Subtraction of vectors

Vectors and Their Properties Example 3.1: Taking a trip A car travels 20 km due north and then 35.0 km in a direction 600 west of north. What is the net effect of the car’s trip?

Components of a Vector Components of a vector A vector in two-dimension can be specified by a pair of coordinates y-component x-component tan-1q defined in (-900,900). Add 1800 when the vector is in 2nd or 3rd quadrant.

Components of a Vector Components of a vector in different coordinates A vector in two-dimension can be specified by a pair of coordinates If a different coordinate system used, the components are different to represent the same vector. y Vector addition by components: By x But use the same coordinate system for both vectors Bx

Components of a Vector Example 3.3: Take a hike An example Example 3.3: Take a hike 1st day : 25.0 km southeast 2nd day : 40.0 km in a direction 60.0o north of east Determine the components of the hiker’s displacements in the 1st and 2nd days.

Displacement, Velocity, & Acceleration in Two-Dimension Displacement in 2D A position vector describes the position of an object at a time. An object’s displacement from ti to tf is defined by:

Displacement, Velocity, & Acceleration in Two-Dimension Velocity in 2D An object’s average velocity during a time interval Dt is: An object’s instantaneous velocity is:

Displacement, Velocity, & Acceleration in Two-Dimension Acceleration in 2D An object’s average acceleration during a time interval Dt is: An object’s instantaneous acceleration is:

Motion in Two-Dimension Motion in 2D: horizontal and vertical direction In this chapter, we will learn movement of an object in both the x- and y-direction simultaneously under constant acceleration. An example: projectile motion under influence of gravity

Motion in Two-Dimension Projectile motion under influence of gravity Let’s examine the motion of an object that is tossed into air but let’s neglect the effects of air resistance and the rotation of Earth. It was experimentally proven that the horizontal and vertical motions are completely independent of each other. Motion in one direction has no effect on motion in the other direction. So, in general, the equations of constant acceleration we learned in Lecture 2 follow separately for both the x-direction and the y-direction.

Motion in Two-Dimension Projectile motion under influence of gravity Let’s assume that at t=0, the projectile leaves the origin with an initial velocity with an angle with the horizontal. x-direction:

Motion in Two-Dimension Projectile motion under influence of gravity Let’s assume that at t=0, the projectile leaves the origin with an initial velocity with an angle with the horizontal. y-direction:

Motion in Two-Dimension Projectile motion under influence of gravity Plug-in all the known quantities. Equations that describe the motion in the x-direction: Equations that describe the motion in the y-direction: Velocity :

Motion in Two-Dimension Projectile motion under influence of gravity Trajectories as a function of the projection angle Note : Given the displacement in x there are two corresponding projection angles. o o o o o

Motion in Two-Dimension Examples Problem 3.5: Stranded explorers (a) What is the range of the package? range of package

Motion in Two-Dimension Examples Problem 3.5: Stranded explorers (b) What are the velocity components of the package at impact? x component: y component: range of package

Motion in Two-Dimension Examples Problem 3.6: The long jump (a) How long does it take for the jumper to reach the max. height? y component: at max.height vy =0 v0= 11.0 m/s q=20.0o

Motion in Two-Dimension Examples Problem 3.6: The long jump (b) Find the maximum height he reaches. y component: From part (a) v0= 11.0 m/s q=20.0o

Motion in Two-Dimension Examples Problem 3.6: The long jump (c) Find the horizontal distance he jumps. Displacement in x: v0= 11.0 m/s q=20.0o

Motion in Two-Dimension Examples Problem 3.8: The rocket (a) Find the rocket’s velocity in y direction. Eq.3.14c: (b) Find the rocket’s velocity in x direction. Eq.3.12a: Eq.3.11a:

Motion in Two-Dimension Examples Problem 3.8: The rocket (c) Find the magnitude and direction of the rocket’s velocity.

Relative Velocity What is relative velocity? The measured velocity of an object depends on the velocity of the observer with respect to the object. Relative velocity relates velocities measured by two different observers, one moving with respect to the other. Measurements of velocity depend on the reference frame of the observer where the reference frame is a just coordinate system used to measure physical quantities such as velocity, acceleration etc. Most of time, we will use a stationary frame of reference, relative to earth, but occasionally we will use a moving frame of reference.

Relative Velocity More elaborate definition: What is relative velocity? (cont’d) More elaborate definition: - Let’s define E as a stationary observer with respect to Earth - A and B as two moving cars

Relative Velocity relative to; with respect to What is relative velocity? (cont’d) relative to; with respect to

Relative Velocity Examples Example 3.10: Crossing a river

Relative Velocity Examples Example 3.10: Crossing a river (cont’d)