CHAPTER 3 VECTOR quantities: Two-imensional Motion and Vectors Vectors ave magnitude and direction. (x, y) Representations: y (x, y) (r, ) x Oter vectors: velocity, acceleration, momentum, force Vector Addition/Subtraction Vector Components 2nd vector begins at end of first vector Order doesn t matter Vector addition Cartesian components are projections along te x- and y-axes A x = Acos A y = Asin Vector subtraction A B can be interpreted as A+(-B) Going backwards, A = A x 2 + A y 2 and = tan "1 A y A x Example 3.1a Example 3.1b Te magnitude of (A-B) is : b) =0 Te x-component of (A-B) is: b) =0
Te y-component of (A-B) > 0 Example 3.1c Example 3.2 Alice and Bob carry a bottle of wine to a picnic site. Alice carries te bottle 5 miles due east, and Bob carries te bottle anoter 10 miles traveling 30 degrees nort of east. Carol, wo is bringing te glasses, takes a sort cut and goes directly to te picnic site. b) =0 How far did Carol walk? Wat was Carol s direction? Carol 14.55 miles, at 20.10 degrees Alice Bob Arcsin, Arccos and Arctan: Watc out v = "r / "t 2-dim Motion: Velocity same sine same cosine same tangent It is a vector (rate of cange of position) Grapically, Trajectory Arcsin, Arccos and Arctan functions can yield wrong angles if x or y are negative. Multiplying/ividing Vectors by Scalars, e.g. "r/"t Vector multiplied/divided by scalar is a vector Magnitude of new vector is magnitude of orginal vector multiplied/divided by scalar irection of new vector same as original vector Principles of 2-d Motion X- and Y-motion are independent Two separate 1-d problems To get trajectory (y vs. x) 1. Solve for x(t) and y(t) 2. Invert one Eq. to get t(x) 3. Insert t(x) into y(t) to get y(x)
Projectile Motion Projectile Motion X-motion is at constant velocity a x =0, v x =constant Acceleration is constant Y-motion is at constant acceleration a y =-g Note: we ave ignored air resistance rotation of eart (Coriolis force) Pop and rop emo Te Ballistic Cart 1. Write down x(t) x =,x t 2. Write down y(t) y =,y t 1 2 gt 2 3. Invert x(t) to find t(x) t = x /,x Finding Trajectory, y(x) 4. Insert t(x) into y(t) to get y(x) y =,y x 1 g x 2 2,x 2,x Trajectory is parabolic Example 3.3 An airplane drops food to two starving unters. Te plane is flying at an altitude of 100 m and wit a velocity of 40.0 m/s. How far aead of te unters sould te plane release te food? 181 m X
Example 3.4a Example 3.4b Te Y-component of v at A is : b) 0 Te Y-component of v at B is b) 0 Example 3.4c Example 3.4d Te Y-component of v at C is: Te speed is greatest at: b) 0 a) A b) B c) C d) Equal at all points Example 3.4e Example 3.4f Te X-component of v is greatest at: a) A b) B c) C d) Equal at all points Te magnitude of te acceleration is greatest at: a) A b) B c) C d) Equal at all points
Range Formula Good for wen y f = y i x = v i,x t y = v i,y t 1 2 gt 2 = 0 t = 2v i,y g x = 2v i,xv i,y g x = v 2 i g sin2" = 2v i 2 cos" sin" g R = v 2 i g sin2 Range Formula Maximum for =45 Example 3.5a Example 3.6 A softball leaves a bat wit an initial velocity of 31.33 m/s. Wat is te maximum distance one could expect te ball to travel? A cannon urls a projectile wic its a target located on a cliff =500 m away in te orizontal direction. Te cannon is pointed 50 degrees above te orizontal and te muzzle velocity is 100 m/s. Find te eigt of te cliff? 100 m 299 m Example 3.7, Soot te Monkey A unter is a distance L = 40 m from a tree in wic a monkey is perced a eigt =20 m above te unter. Te unter soots an arrow at te monkey. However, tis is a smart monkey wo lets go of te branc te instant e sees te unter release te arrow. Te initial velocity of te arrow is v = 50 m/s. Must find,y /v x in terms of and L 1. Heigt of arrow y arrow =,y t 1 2 gt 2 2. Heigt of monkey y monkey = 1 2 gt 2 Solution: A. If te arrow traveled wit infinite speed on a straigt line trajectory, at wat angle sould te unter aim te arrow relative to te ground? =Arctan(/L)=26.6 B. Considering te effects of gravity, at wat angle sould te unter aim te arrow relative to te ground? 3. Require monkey and arrow to be at same place 1 2 gt 2 =,y t 1 2 gt 2 =,y t =,y L v x,,y v x = L Aim directly at Monkey
Soot te Monkey emo Relative velocity Velocity always defined relative to reference frame. All velocities are relative Relative velocities are calculated by vector addition/ subtraction. Acceleration is independent of reference frame For ig v ~c, rules are more complicated (Einstein) Example 3.8 Relative velocity in 2-d A plane tat is capable of traveling 200 m.p.. flies 100 miles into a 50 m.p.. wind, ten flies back wit a 50 m.p.. tail wind. How long does te trip take? Wat is te average speed of te plane for te trip? Sum velocities as vectors velocity relative to ground = velocity relative to medium + velocity of medium. v be = v br + v re 1.067 ours = 1 r. and 4 minutes 187.4 mp Boat wrt eart boat wrt river river wrt eart 2 Cases Example 3.9 An airplane is capable of moving 200 mp in still air. Te plane points directly east, but a 50 mp wind from te nort distorts is course. Wat is te resulting ground speed? Wat direction does te plane fly relative to te ground? pointed perpendicular to stream travels perpendicular to stream 206.2 mp 14.0 deg. sout of east
Example 3.10 An airplane is capable of moving 200 mp in still air. A wind blows directly from te Nort at 50 mp. Te airplane accounts for te wind (by pointing te plane somewat into te wind) and flies directly east relative to te ground. Wat is te plane s resulting ground speed? In wat direction is te nose of te plane pointed? 193.6 mp 14.5 deg. nort of east Example 3.11a Tree airplanes, A, B and C, wit identical air speeds fly from Williamston, MI, towards Tallaassee, FL, wic is directly sout. A flies on Monday wen tere is a strong wind from te west. A aims te plane sout but is blown off course. B also leaves Monday, but aims a bit into te wind and lands in Tallaassee. C flies on Tuesday, a calm and windless day, and flies directly to Tallaassee. Wic plane(s) as(ave) te HIGHEST ground speed? A) A B) B C) C ) A and B E) B and C Example 3.11b Tree airplanes, A, B and C, wit identical air speeds fly from Williamston, MI, towards Tallaassee, FL, wic is directly sout. A flies on Monday wen tere is a strong wind from te west. A aims te plane sout but is blown off course. B also leaves Monday, but aims a bit into te wind and lands in Tallaassee. C flies on Tuesday, a calm and windless day, and flies directly to Tallaassee. Wic plane(s) as(ave) te LOWEST ground speed? A) A B) B C) C ) A and B E) B and C