 # Vectors and Two-Dimensional Motion

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Vectors and Two-Dimensional Motion
Chapter 3 Vectors and Two-Dimensional Motion

Vectors Vectors – physical quantities having both magnitude and direction 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

A + B = B + A

(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

In 2D case:

Chapter 3 Problem 14 A quarterback takes the ball from the line of scrimmage, runs backwards for 10.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 50.0-yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement?

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

Displacement The displacement of the object 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

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 58 A 2.00-m-tall basketball player is standing on the floor 10.0 m from the basket. If he shoots the ball at a 40.0° angle with the horizontal, at what initial speed must he throw the basketball so that it goes through the hoop without striking the backboard? The height of the basket is 3.05 m.

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?

Chapter 3 Problem 2: Approximately 484 km (b) Approximately 18.1° N of W

Chapter 3 Problem 6: Approximately 6.1 units at 113° (b) Approximately 15 units at 23°