Chapter 3: Two-Dimensional Motion and Vectors Objectives Define vectors and scalars. Understand simple vector operations like addition, subtraction, and multiplication. Be able to do vectors operations graphically. Vectors and Scalars vector quantity: has both magnitude and direction (e.g.: velocity, acceleration) scalar quantity: has magnitude only (e.g.: distance, speed) Arrows are useful when working with vectors quantities. 30 m/s Vector Addition in 1-D boat = 7.5 m/s current = 2.2 m/s 7.5 m/s m/s resultant vector = 9.7 m/s boat = 7.5 m/s 7.5 m/s + (-2.2 m/s) resultant = 5.3 m/s current = m/s Vector Addition in 2-D 120 mph 90 mph Add tail-to-tipto find the resultant. resultant A plane traveling north at 120 mph is blown eastward by a 90 mph wind. Vector Addition Vectors can be moved horizontally to themselves and can be added in any order. Same resultant vector! Vector Subtraction, Multiplication To subtract a vector, add its opposite. Multiplying or dividing vectors by scalars results in vectors. A 3A A+B A-B B 0.5 B Adding Vectors Graphically A = 55 m/s north (0 o ), B = 35 m/s west (270 o ) A + B = ? C = 310 mph at 45 o, D = 60 mph at 180 o C + 2.5D = ? D C = ? Objectives Be able to use trigonometry to solve vector addition problems. Be able to use trigonometry to do vector resolution. Vector Addition Problem A car goes 12.5 km east, turns right and travels 18.3 km south, and then turns left and goes another 16.6 km east. What is the displacement of the car? Include the direction! Vector Resolution vector resolution: the process of determining the components of a vectorusually involves trigonometry velocity of a flying ball vertical component horizontal component Vector Resolution Problem = 28.0 o A rocket is launched at 465 m/s at an angle 28.0 o above the horizon. What are the initial v x and v y components? vxvx vyvy v = 465 m/s Objectives Understand the independence of vertical and horizontal motion. Discuss the path of a projectile. Solve projectile problems for values such as maximum height, range, and time. Projectile Motion What hits the ground first, a dropped ball or a ball that is projected horizontally from the same height? They hit at the same time. Considering relative motion, the balls are really doing the same thing (falling). Horizontal and vertical motion are independent; the only thing in common to each is time. horizontal: constant speed, v x = d/t vertical: uniform acceleration, use kinematic equations Projectile Motion Cannonball moves at a constant speed horizontally. Cannonball accelerates downward at g. Projectile Motion Projectile Problems A marble rolling at 0.65 m/s falls off a table that is 1.03 m high. What is the range? (1) find the time from the height (2) find the range from the time Objects Projected at an Angle You must determine initial speed components. Time is still the common variable. A cannonball is launched at 125 m/s at a 32 o angle. What is the range? Objects Launched at an Angle -5m -20 m -45 m A projectile falls below the straight path the same distance it would fall if dropped from rest. 1 s 2 s 3 s Path of a Projectile Ignoring air resistance, projectiles follow a parabolic path. In reality, air resistance alters the actual path. What launching angle results in the greatest range? 45 o Satellites satellite: a projectile with enough horizontal velocity (> 8 km/s or mph) that its curved path matches the curvature of the earth. 8 km 5 m Satellites Satellite paths (orbits) are actually elliptical. These are all satellites! Newtonian Mountain