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Transcript
Two-Dimensional Motion and Vectors Honors Physics Unit 2 • Our last unit focused on motion in a straight line (usually forward and backward) using + and – signs to indicate direction • This unit focuses on motion that is not in a straight line (Re)Introduction to Vectors • Vectors indicate direction, Scalars do not **Symbol Change Alert** • Sometimes we use the same symbol to represent 2 different things. (velocity and speed = v) Velocity is a vector quantity and Speed is scalar • Your book will put vector quantities in bold (v) and scalar quantities in italics (v) • You can distinguish between the two when you write them with an arrow above the symbol.( ) Vectors • Remember: In diagrams, vectors are shown as arrows that point in the direction of the vector. – The length of the vector arrow is proportional to it’s magnitude Resultants • The sum of two or more vectors is called the Resultant. – make sure that you have the same units when adding vectors and that you are describing similar quantities (ex. You wouldn’t add velocity and displacement together, even though they are both vector quantities. You wouldn’t add feet and meters together either) Adding Vectors Graphically • If you walked 1600m (a) to a friends house, then 1600m (b) to school, what is your total displacement You must draw the graph to scale. For example, use 1cm equals 50m Using a ruler, you can measure the length of the resultant, and multiply the length by the scale used to get it’s magnitude. Adding Vector Quantities • Using a protractor in the last example can give you a direction of the resultant. • Measure the angle between the resultant and the first vector Remember Me? • Head to tail method: • Head to tail method: (More than 2 forces) R Head to Tail • Vectors can be moved (added) in any order Homework: Problems 1-5 Page 85; Holt Physics Determining Resultant Magnitude and Direction • For homework, we found magnitude and direction using graphs – Time consuming – Accuracy • There is a simpler way.. Magnitude of a resultant • Pythagorean theorem For any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, or legs c2 = a2 + b2 y d Δy Δx x Direction of a Resultant • Tangent Function For a right triangle the tangent of an angle is the ratio of the opposite and adjacent sides(legs) • It is knowing the angle that is important to determining direction, so we can use the inverse of the tangent function Example Problem • You climb on of Egypt’s pyramids that has a height of 136m and a width of 2.30E2m. What is the magnitude and direction of the displacement for you as you climb from the bottom to the top? Given: Unknowns: Δy=136m d=? Δx = 115m θ=? • Use the Pythagorean theorem to find the magnitude of your displacement d2 = 1152 + 1362 d = 178m • Now for direction use inverse tangent θ=49.8° More Practice • A girl delivering newspapers travels three blocks west, four blocks north and then six blocks east. 1. What is her resultant displacement? 2. What is the total distance she travels More Practice • A quarterback takes the ball from the line of scrimmage, runs backward for 10.0 yards, and then runs sideways for 15.0 yards. At this point, he throws a 50.0 yard pass straight down the field. What is the magnitude of the football’s resultant displacement? More Practice • A shopper pushes a cart 40.0m south down one aisle and then turns 90.0° and moves 15.0m. He then makes another 90.0° turn and moves 20.0m. Find the shopper’s total displacement. (there can be more than one correct answer) Resolving Vectors into Components • What do we mean by “components”? – Horizontal part (x-axis) and Vertical part (y-axis) y x If given the angle and the magnitude of the vector force, its components can be found using trigonometry Vector Components Example • Find the components of the velocity of a helicopter traveling 95km/h at an angle of 35° y to the ground. = 95km/h Given: = 95km/h θ = 35° Unknowns: Vx ; Vy Step 1: Draw Diagram Step 2: Use Trig to solve Step 3: Check your work Sin 35 = Vy/ Vy 35° x Cos 35 = Vx/ Vx Vy = 54 km/h Vx = 78 km/h 782 + 542 = 952 ?? Practice Problems • How fast must a truck travel to stay beneath an airplane that is moving 105km/h at an angle of 25° to the ground? • What is the magnitude of the vertical component of the velocity of the plane? Two More • A truck drives up a hill with a 15° incline. If the truck has a constant speed of 22m/s, what are the horizontal and vertical components of the truck’s velocity? • What are the horizontal and vertical components of a cat’s displacement when the cat has climbed 5m directly up a tree? Non Perpendicular Vectors Homework • Problem Workbook: Pg 18 (1-7) & Pg 22 (1-5) Projectiles • A projectile is an object that moves through air and space with only gravitation force acting on it. • Projectiles have a horizontal and a vertical component • Horizontal component remains constant as long as no other horizontal force is acting on it Projectile • The Vertical component has the same properties of an object in free fall. (a constant acceleration due to gravity) • The two components (horizontal and vertical) are independent of each other. – Gravity has no effect on the horizontal velocity and the horizontal velocity has no effect on the acceleration due to gravity Projectiles • Projectiles launched at any Angle will fall vertically from the ideal straight line, equally – Launched Horizontally (Angle= 0°) – Launched Above the Horizontal – Launched Below the Horizontal Projectiles • Horizontal Projectile 5m 20 m 45 m Projectiles • Launched Above the Horizontal 5m 20 m 45 m Projectiles • Where did the 5m, 20m and 45m distances come from? – Gravity acting on the object • Remember the vertical distance that an object falls beneath the ideal straight line is the same as the distance it would travel in a free fall So, we can use the distance equation we had for objects in a freefall Δy = ½(g)t2 Since ‘g’ is constant (10.0m/s2) the equation can be simplified to Δy = 5t2 Projectiles • The path that projectiles follow is called a parabola. The angle of the launch determines the shape of the parabola Projectiles • You can find the Vertical and Horizontal components of the object at any time (t) which will let you solve for a resultant vector velocity. • Ex) Stude tosses a ball into the air with a horizontal component of 5m/s and a vertical component of 30m/s. In 1 second intervals, find the velocity components as the ball travels up and the resultant velocities as the ball travels downward So, what about that frog… • Givens: Δt = 0.60s v = 5.0m/s Δx = 2.4m • Unknown: θ 1. Find the horizontal velocity needed to leap 2.4m in 0.60s. V = Δx/Δt = 4.0m/s 5.0m/s 2.4m 2. Using the horizontal velocity and the resultant velocity calculate the angle needed 5.0m/s Cos-1 (4.0/5.0) = 36.8° 4.0m/s