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

Two-Dimensional Motion and Vectors Scalar and Vector Quantities Scalar Quantity Magnitude (number and units) ex. 15km, 250C Vector Quantity Magnitude and Direction ex. 15km north

Two-Dimensional Motion and Vectors Represented by Arrows Direction of the Arrow Represents the Direction of the Vector The Length of the Arrow is Proportional to the Magnitude of the Vector

Two-Dimensional Motion and Vectors Components Two Perpendicular Components Add to Form Resultant (Displacement) A Vector May be Broken Down Into Its “X” and “Y” Components Resultant Vector Sum of Two or More Vectors

Two-Dimensional Motion and Vectors Components Vector “r” West Component North Component

Two-Dimensional Motion and Vectors Components

Two-Dimensional Motion and Vectors Components

Two-Dimensional Motion and Vectors Ex. Height Determination Distance to Cliff = 500’ Angle = 34o

Two-Dimensional Motion and Vectors Ex. Height Determination Distance to Cliff = 500’ Angle = 34o

Two-Dimensional Motion and Vectors Components Determination of Direction Observe Direction From Tail of Resultant to Right Angle Formed by X and Y Components

Two-Dimensional Motion and Vectors Determination of Direction

Two-Dimensional Motion and Vectors Adding and Subtracting Vectors Adding Vectors Graphically To add vectors A and B, place the tail of B at the head of A. The sum, C = A+B, is the vector extending from the tail of A to the head of B.

Two-Dimensional Motion and Vectors Adding and Subtracting Vectors Adding Vectors Graphically

Two-Dimensional Motion and Vectors Adding and Subtracting Vectors Adding Vectors Graphically C = A + B

Two-Dimensional Motion and Vectors Adding and Subtracting Vectors Adding Vectors Graphically C = A + B C = B + A

Two-Dimensional Motion and Vectors Adding and Subtracting Vectors Adding Vectors Using Components

Two-Dimensional Motion and Vectors Adding and Subtracting Vectors Adding Vectors Using Components A=5.00m B=4.00m

Two-Dimensional Motion and Vectors Adding and Subtracting Vectors Adding Vectors Using Components A=5.00m B=4.00m

Two-Dimensional Motion and Vectors Adding and Subtracting Vectors Adding Vectors Using Components A=5.00m B=4.00m Ax=2.50m Ay=4.33m Bx=3.76m By=1.37m

Two-Dimensional Motion and Vectors Adding and Subtracting Vectors Adding Vectors Using Components

Two-Dimensional Motion and Vectors Adding and Subtracting Vectors Subtracting Vectors D = A - B D = A + (-B) Now the same rules for adding apply 

Two-Dimensional Motion and Vectors Horizontal and Vertical Motions are Independent

Two-Dimensional Motion and Vectors An Object with a Horizontal Velocity will Continue with the Same Velocity in the Horizontal Direction Even While an Independent Vertical Motion Acts on the Object Each Motion Continues as if the Other Motion Were Not Present

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Velocity vi = 26 m/s What is the turtle’s position after 5.0s?

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Velocity vi = 26 m/s t = 5.0s

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Velocity vi = 26 m/s t = 5.0s What are the x and y components of the vector?

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Velocity vi = 26 m/s t = 5.0s q = 25o

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Velocity vi = 26 m/s What is the turtle’s displacement after 5.0s?

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Velocity vi = 26 m/s What is the turtle’s displacement after 5.0s? Well, We Could Find the x and y Components First.

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Velocity vi = 26 m/s t = 5.0s

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Velocity vi = 26 m/s t = 5.0s vix = 23.56m/s viy = 10.99m/s

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Acceleration Remember this? We Can use This on Each Component

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Acceleration And Remember This? We Can use This on Each Component Too

Two-Dimensional Motion and Vectors Motion in Two Dimensions Constant Acceleration And Remember This? We Can use This on Each Component Too

Two-Dimensional Motion and Vectors Problem As you walk to class with a constant speed of 1.60 m/s, you are moving in a direction that is 15.0° north of east. How long does it take you to move 30.0 m north?

Two-Dimensional Motion and Vectors Solution v = 1.60m/s q = 15.0o y = 30.0m t = ?

Two-Dimensional Motion and Vectors Problem

Two-Dimensional Motion and Vectors Problem Two canoeists start paddling at the same time and head toward a small island in a lake. Canoeist 1 paddles with a speed of 1.35m/s at an angle of 45° north of east. Canoeist 2 starts on the opposite shore of the lake, a distance of 1.5 km due east of canoeist 1. In what direction relative to north must canoeist 2 paddle to reach the island?

Two-Dimensional Motion and Vectors Solution Canoeist 1’s 45° path determines an isosceles right triangle whose legs measure 1.0 km. So canoeist 2’s path determines a right triangle whose legs measure 1.0 km and 0.5 km. Then for canoeist 2…

Two-Dimensional Motion and Vectors Problem Two canoeists start paddling at the same time and head toward a small island in a lake. Canoeist 1 paddles with a speed of 1.35m/s at an angle of 45° north of east. Canoeist 2 starts on the opposite shore of the lake, a distance of 1.5 km due east of canoeist 1. What speed must canoeist 2 have if the two canoes are to arrive at the island at the same time?

Two-Dimensional Motion and Vectors Solution v1 = 1.35m/s d1 = d2 = q1 = 45o v2 = ?

Two-Dimensional Motion and Vectors Homework Pages 113 - 114 Problems 6 (a, 5.00 units @ 53.1o below x-axis b, 5.00 units @ 53.1o above x-axis c, 8.54 units @ 69.4o below x-axis d, 5.00 units @ 127o clockwise from positive x-axis) 9 (15.3m @ 58.4o S of E) 27 (2.81km E, 1.31km N) 29 (240.0m @ 57.23o S of W)

Two-Dimensional Motion and Vectors Projectile Motion A Projectile is an Object that is Launched into Motion by Any Means and Then Allowed to Follow a Path Determined Solely by the Influence of Gravity

Two-Dimensional Motion and Vectors Projectile Motion Assumptions… Ignore Air Resistance g = -9.81m/s2 Ignore the Earth’s Rotation

Two-Dimensional Motion and Vectors Projectile Motion Free Fall /w Initial Horizontal Velocity Parabolic Path

Two-Dimensional Motion and Vectors Projectile Motion Acceleration Horizontal Acceleration is Always Zero (Constant Velocity) Vertical Acceleration is Always –g

Two-Dimensional Motion and Vectors Projectile Motion Equations ax = 0, ay = -g Displacement

Two-Dimensional Motion and Vectors Projectile Motion Equations ax = 0, ay = -g Velocity

Two-Dimensional Motion and Vectors Projectile Motion Zero Launch Angle

Two-Dimensional Motion and Vectors Projectile Motion Zero Launch Angle q = 0o

Two-Dimensional Motion and Vectors Projectile Motion Zero Launch Angle Where Will an Object Land? y = 0 = ground h = release height

Two-Dimensional Motion and Vectors Projectile Motion

Two-Dimensional Motion and Vectors Projectile Motion General Launch Angle

Two-Dimensional Motion and Vectors Projectile Motion General Launch Angle

Two-Dimensional Motion and Vectors Projectile Motion Time of Flight Full At Specific dx

Two-Dimensional Motion and Vectors Projectile Motion Maximum Range (R) Projectile Range is Dependant on the Angle q that the Projectile is Fired from Horizontal

Two-Dimensional Motion and Vectors Projectile Motion Maximum Range (R) With Air Resistance

Two-Dimensional Motion and Vectors Projectile Motion Maximum Height (ymax)

Two-Dimensional Motion and Vectors Projectile Motion Symmetry of Projectile Motion Parabolic Flight Peak of Flight = ½ Time of Flight Vertical Impact Velocity = -Vertical Launch Velocity Impact Angle = Launch Angle

Two-Dimensional Motion and Vectors Problem What is the acceleration of a projectile when it reaches its highest point?

Two-Dimensional Motion and Vectors Solution The only acceleration on a projectile is gravity. At any point, the acceleration on a projectile is equal to g.

Two-Dimensional Motion and Vectors Problem A particle passes through the origin with a velocity of 6.2m/s in the y axis. If the particle’s acceleration is -4.4m/s2 in the x axis what are its x and y positions after 5.0 s?

Two-Dimensional Motion and Vectors Solution ax = -4.4m/s2 vy = 6.2m/s t = 5.0s x = ? y = ?

Two-Dimensional Motion and Vectors Problem Playing shortstop, you pick up a ground ball and throw it to second base. The ball is thrown horizontally, with a speed of 22m/s, directly toward point A. When the ball reaches the second baseman 0.45 s later, it is caught at point B. How far were you from the second baseman and what is the distance of the vertical drop?

Two-Dimensional Motion and Vectors Solution vx = 22m/s t = 0.45s

Two-Dimensional Motion and Vectors Problem A soccer ball is kicked with a speed of 9.50m/s at an angle of 25.0° above the horizontal. If the ball lands at the same level from which it was kicked, how long was it in the air?

Two-Dimensional Motion and Vectors At the peak of the ball’s trajectory, Solution so The total time is twice this. 2t = 2(0.40926 s) = 0.819 s

Two-Dimensional Motion and Vectors Problem A golfer gives a ball a maximum initial speed of 30.0m/s. What is the highest tree the ball could clear on its way to the hole-in-one?

Two-Dimensional Motion and Vectors Solution vi = 30.0m/s q = 45o ymax = ?

Two-Dimensional Motion and Vectors Problem

Two-Dimensional Motion and Vectors Problem Astronomers have discovered several volcanoes on Io, a moon of Jupiter. One of them, named Loki, ejects lava to a maximum height of 2.00x105m. What is the initial speed of the lava? (The acceleration of gravity on Io is 1.80m/s2)

Two-Dimensional Motion and Vectors Solution ymax = 2.00x105m/s g = 1.80m/s2 At the lava’s maximum height,

Two-Dimensional Motion and Vectors Problem A ball thrown straight upward returns to its original level in 2.50s. A second ball is thrown at an angle of 40.0° above the horizontal. What is the initial speed of the second ball if it also returns to its original level in 2.50s?

Two-Dimensional Motion and Vectors Solution t1 = 2.50s t2 = 2.50s q1 = 90o q2 = 40.0o vo = ? at the ball’s maximum height.

Two-Dimensional Motion and Vectors Relative Motion Solved by the use of Vector Addition and Subtraction

Two-Dimensional Motion and Vectors Relative Motion

Two-Dimensional Motion and Vectors Relative Motion?

Two-Dimensional Motion and Vectors Relative Motion Climbing ladder @ 0.20 m/s (vpt) Train speed is 0.70 m/s (vtg) What is the Relative Velocity of the Person on the Train? (vpg)

Two-Dimensional Motion and Vectors Relative Motion vpt = 0.20 m/s vtg = 0.70 m/s vpg = ?

Two-Dimensional Motion and Vectors Problem You are driving up a long inclined road. After 1.7 miles you notice that signs along the roadside indicate that your elevation has increased by 550 ft. What is the angle of the road above the horizontal?

Two-Dimensional Motion and Vectors Solution

Two-Dimensional Motion and Vectors Problem You are driving up a long inclined road. After 1.7 miles you notice that signs along the roadside indicate that your elevation has increased by 550 ft. How far do you have to drive to gain an additional 150 ft of elevation?

Two-Dimensional Motion and Vectors Solution

Two-Dimensional Motion and Vectors Problem A lighthouse that rises 49 ft above the surface of the water sits on a rocky cliff that extends 19 ft from its base. A sailor on the deck of a ship sights the top of the lighthouse at an angle of 30.0° above the horizontal. If the sailor’s eye level is 14 ft above the water, how far is the ship from the rocks?

Two-Dimensional Motion and Vectors Solution h = 49’ r = 19’ s = 14’ d = ? y = h – s x = d + r

Two-Dimensional Motion and Vectors Problem The x and y components of a vector r are rx=14m and ry= -9.5m respectively. If both rx and ry are doubled, how do the previous answers change?

Two-Dimensional Motion and Vectors Solution The direction does not change. The magnitude is doubled.

Two-Dimensional Motion and Vectors Homework Pages 114 - 116 Problems 35 (a, 2.77x105m b, 284s) 39 (10.8m) 41 (80m; 210m) 49 (a, 14.1o N of W b, 199km/hr) 51 (a, 23.2o upstream b, 8.72m/s across)