Vectors 2D & 3D Motion Lecturer: Professor Stephen T. Thornt on

Given that A + B = C, and that lAl + lBl = lCl, how are vectors A and B oriented with respect to each other? A) they are perpendicular to each other B) they are parallel and in the same direction C) they are parallel but in the opposite direction D) they are at 45° to each other E) they can be at any angle to each other Reading Quiz I

Given that A + B = C, and that lAl + lBl = lCl, how are vectors A and B oriented with respect to each other? A) they are perpendicular to each other B) they are parallel and in the same direction C) they are parallel but in the opposite direction D) they are at 45° to each other E) they can be at any angle to each other The only time vector magnitudes will simply add together is when the direction does not have to be taken into account (i.e., the direction is the same for both vectors). In that case, there is no angle between them to worry about, so vectors A and B must be pointing in the same direction. Reading Quiz

Last Time Problem solving Motion with constant acceleration Free fall

Today More conceptual quizzes Vectors Two and three dimensional motion

How to Succeed in Physics 1)Don’t get behind. Don’t wait until the last evening to start homework! 2)When you can’t work problems, it usually means you don’t understand the concepts or have difficulty with the mathematics. 3)Work lots of problems that have answers at the back of the book. 4)Try the questions at the end of the chapter. 5)Don’t get behind!

Falling Water Balloons. Roger sees water balloons fall past his window. He notices that each balloon strikes the sidewalk 0.83 s after passing his window. Roger’s room is on the third floor, 15 m above the sidewalk. (a) How fast are the balloons traveling when they pass Roger’s window? (b) Assuming the balloons are being released from rest, from what floor are they being released? Each floor of the dorm is 5.0 m high.

A ball is thrown straight upward with some initial speed. When it reaches the top of its flight (at a height h), a second ball is thrown straight upward with the same initial speed. Where will the balls cross paths? A) at height h B) above height h/2 C) at height h/2 D) below height h/2 but above 0 E) at height 0 Conceptual Quiz

A ball is thrown straight upward with some initial speed. When it reaches the top of its flight (at a height h), a second ball is thrown straight upward with the same initial speed. Where will the balls cross paths? A) at height h B) above height h/2 C) at height h/2 D) below height h/2 but above 0 E) at height 0 The first ball starts at the top with no initial speed. The second ball starts at the bottom with a large initial speed. Because the balls travel the same time until they meet, the second ball will cover more distance in that time, which will carry it over the halfway point before the first ball can reach it. Conceptual Quiz

t x A) it speeds up all the time B) it slows down all the time C) it moves at constant velocity D) sometimes it speeds up and sometimes it slows down E) not really sure The graph of position vs. time for a car is given below. What can you say about the velocity of the car over time? The graph of position vs. time for a car is given below. What can you say about the velocity of the car over time? Conceptual Quiz

A) it speeds up all the time B) it slows down all the time C) it moves at constant velocity D) sometimes it speeds up and sometimes it slows down E) not really sure t x The graph of position vs. time for a car is given below. What can you say about the velocity of the car over time? The graph of position vs. time for a car is given below. What can you say about the velocity of the car over time? The car slows down all the time because the slope of the x vs. t graph is diminishing as time goes on. Remember that the slope of x vs. t is the velocity! At large t, the value of the position x does not change, indicating that the car must be at rest. Conceptual Quiz

Consider the line labeled B in the v vs. t plot. How does the speed change with time for line B? Consider the line labeled B in the v vs. t plot. How does the speed change with time for line B? v t A B A) decreases B) increases C) stays constant D) increases, then decreases E) decreases, then increases Conceptual Quiz

Consider the line labeled B in the v vs. t plot. How does the speed change with time for line B? Consider the line labeled B in the v vs. t plot. How does the speed change with time for line B? v t A B A) decreases B) increases C) stays constant D) increases, then decreases E) decreases, then increases In case B, the initial velocity is positive but the magnitude of the velocity decreases toward zero. After this, the magnitude increases again, but becomes negative, indicating that the object has changed direction. Conceptual Quiz

v t A v t B v t C v t D You drop a very bouncy rubber ball. It falls, and then it hits the floor and bounces right back up to you. Which represents the velocity v vs. t graph for this motion? You drop a very bouncy rubber ball. It falls, and then it hits the floor and bounces right back up to you. Which represents the velocity v vs. t graph for this motion?

v t A v t B v t C v t D You drop a very bouncy rubber ball. It falls, and then it hits the floor and bounces right back up to you. Which of the following represents the velocity v vs. t graph for this motion? You drop a very bouncy rubber ball. It falls, and then it hits the floor and bounces right back up to you. Which of the following represents the velocity v vs. t graph for this motion? negative increasing positive decreasing Initially, the ball is falling down, so its velocity must be negative (if UP is positive). Its velocity is also increasing in magnitude as it falls. Once it bounces, it changes direction and then has a positive velocity, which is also decreasing as the ball moves upward. Conceptual Quiz

Falling Ball. A ball is dropped from the top of a 50.0-m-high cliff. At the same time, a carefully aimed stone is thrown straight up from the bottom of the cliff with a speed of 24.0 m/s. The stone and ball collide part way up. How far above the base of the cliff does this happen?

Review of Vectors A scalar is a number with units. It can be positive, negative, or zero. A vector has both a magnitude and direction. When writing a vector we will put an arrow over it. Sometimes a vector is in boldface.

Directions to the library 3 blocks west, 3 blocks north.

Conceptual Quiz A) Between 0 0 and B) Between 90 0 and C) Between and D) Between and E) Cannot be determined with information given.

Answer: C y C) Between and x 

The Sum of Two Vectors We can add vectors.

Can we add vectors either way? (commutative)

Doesn’t matter:

Component Addition of Vectors

Vector Subtraction We can subtract vectors. The minus sign just changes the direction.

Vector Subtraction We can add negative vectors. Head Tail

Which of the following are equal vectors? A) B) C) D) E) Conceptual Quiz

Answer: If we look closely, we see they have the same magnitude and with the minus sign, they have the same direction. C)

Unit Vectors

Multiplying a Vector by a Scalar We can multiply a vector by a scalar.

Vector Component Use Unit vectors make vector addition and subtraction reasonably easy.

Vector Components Draw this new vector. Find magnitude and angle of this vector. Special Extra Problem 3 on Vectors.

Good review of vector use: vectors/vectors.html vectors/vectors.html

Vector Addition & Subtraction. For the vectors given in the figure, determine.

If each component of a vector is doubled, what happens to the angle of that vector? A) it doubles B) it increases, but by less than double C) it does not change D) it is reduced by half E) it decreases, but not as much as half Conceptual Quiz

If each component of a vector is doubled, what happens to the angle of that vector? A) it doubles B) it increases, but by less than double C) it does not change D) it is reduced by half E) it decreases, but not as much as half The magnitude of the vector clearly doubles if each of its components is doubled. But the angle of the vector is given by tan  = 2y/2x, which is the same as tan  = y/x (the original angle). Conceptual Quiz

If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B ? A) same magnitude, but can be in any direction B) same magnitude, but must be in the same direction C) same magnitude, but must be in opposite directions D) different magnitudes, but must be in the same direction E) different magnitudes, but must be in opposite directions Conceptual Quiz

If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B ? A) same magnitude, but can be in any direction B) same magnitude, but must be in the same direction C) same magnitude, but must be in opposite directions D) different magnitudes, but must be in the same direction E) different magnitudes, but must be in opposite directions The magnitudes must be the same, but one vector must be pointing in the opposite direction of the other in order for the sum to come out to zero. You can prove this with the tip-to-tail method. Conceptual Quiz

A certain vector has x and y components that are equal in magnitude. Which of the following is a possible angle for this vector in a standard x-y coordinate system? A) 30° B) 180° C) 90° D) 60° E) 45° Conceptual Quiz

A certain vector has x and y components that are equal in magnitude. Which of the following is a possible angle for this vector in a standard x-y coordinate system? A) 30° B) 180° C) 90° D) 60° E) 45° The angle of the vector is given by tan  = y/x. Thus, tan  = 1 in this case if x and y are equal, which means that the angle must be 45°. Conceptual Quiz