Physics 231. Topic 3: Vectors and two dimensional motion. Alex Brown September MSU Physics 231 Fall

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1 Physics 231 Topic 3: Vectors and two dimensional motion Alex Brown September MSU Physics 231 Fall

2 What s up? (Monday Sept 14) 1) Homework set 01 due Tuesday Sept 15 th 10 pm 2) Learning Resource Center (LRC) in 1248 BPS Monday 9 am to 1pm and 3 pm to 5 pm Tuesday 9 am to 9pm Friday 9am to 1 pm MSU Physics 231 Fall

3 Key Concepts: 2D Motion Vectors and Scalars Two Dimensional Motion Velocity in 2D Acceleration in 2D Projectile motion Throwing a ball or cannon fire Uniform Circular Motion Centripetal acceleration Covers chapter 3 in Rex & Wolfson MSU Physics 231 Fall

4 Extra Credit Quiz You throw a ball straight up (positive direction) and catch it again at the same location where you released it. At the highest point the acceleration is: a) Impossible to tell b) -9.8 m/s 2 c) 9.8 m/s 2 d) 0 MSU Physics 231 Fall

5 Trigonometry a c SOH CAH - TOA: sin =opposite/hypotenuse =a/c cos =adjacent/hypotenuse =b/c tan =opposite/adjacent =a/b Pythagorean theorem: c 2 a 2 b 2 b Note that sin,cos,tan are dimensionless. 2 radians corresponds to 360 o MSU Physics 231 Fall

6 Be Careful. y h y y= hsin() = h cos() Always check carefully which angle is given MSU Physics 231 Fall

7 Vectors and Scalars Scalar: A quantity specified by its magnitude only Vector: A quantity specified both by its magnitude and direction. To distinguish a vector from a scalar quantity, it is usually written with an arrow above it, or in bold to distinguish it from a scalar. Scalar: A Vector: A or A (bold face) MSU Physics 231 Fall

8 Vectors and Scalars MSU Physics 231 Fall

9 Question Are these two vectors the same? Are the lengths of these two vectors the same? No Yes Two vectors are equal if both their lengths and directions are the same! MSU Physics 231 Fall

10 Vector Addition A+B B A B B+A A+B=B+A A MSU Physics 231 Fall

11 MSU Physics 231 Fall Vector operations in equations A B A+B y x (x a,y a ) (x b,y b ) (x a+b,y a+b ) b a b a b b a a b a b a Y Y X X Y X Y X Y X b a b a Y X Example: A+B

12 MSU Physics 231 Fall A B A-B = A + (-B) Vector Subtraction -B b a b a b b a a b a b a Y Y X X Y X Y X Y X

13 MSU Physics 231 Fall A -B A-B=A+(-B) B Vector Subtraction b a b a b b a a b a b a Y Y X X Y X Y X Y X

14 More on Coordinate Systems y b r Cartesian Coordinates: (x,y)=(a,b) Also, r = a î + b ĵ î unit vector in x direction ĵ unit vector in y direction a x Plane Polar Coordinates (r,) 2 2 r a b tan( ) b / a MSU Physics 231 Fall

15 Vector length and its components Y (x a,y a ) Length of vector (use pythagorean theorem): x y a a l tan cos l sin y a / x a x l 2 2 x a y a MSU Physics 231 Fall

16 Vector operations in equations MSU Physics 231 Fall

17 Vector operations in equations ff MSU Physics 231 Fall

18 Question A man walks 5 km/h. He travels 12 minutes to the east, 30 minutes to the south-east and 36 minutes to the north. A) What is the displacement of the man when he s done? B) What is the total distance he walked? A) X Y X 1 Y 1 X Y 2 2 X Y D km 2.5 km =-45 o 3 km Y 2 =2.5 sin()=-1.77 B) = 6.5 km x 2 =2.5 cos()=1.77 MSU Physics 231 Fall

19 Boat crossing the river Flow MSU Physics 231 Fall

20 Question A boat is trying to cross a 1 km wide river in the shortest way (straight across). Its maximum speed (in still water) is 10 km/h. The river is flowing (south) with 5 km/h. 1) At what angle does the captain have to steer the boat the go straight across? 2) How long does it take for the boat to cross the river? MSU Physics 231 Fall

21 Answer A boat is trying to cross a 1 km wide river in the shortest way (straight across). Its maximum speed (in still water) is 10 km/h. The river is flowing with 5 km/h. 1) At what angle does the captain have to steer the boat the go straight across? sin = opposite/hypotenuse = 5/10 = 0.5 = sin -1 (0.5) = 30 o Flow=5km/h 2) How long does it take the boat to cross the river? velocity hor =sqrt(100-25) = 8.66 km/h Time = (1 km)/(8.66 km/h) = h = 6.9 min v x t MSU Physics 231 Fall

22 3) If it doesn t matter at what point the boat reaches the other side, at what angle should the captain steer to cross in the fastest way? 0 o :the horizontal component of the velocity is then maximum. MSU Physics 231 Fall

23 plane in the wind MSU Physics 231 Fall

24 Displacement in 2D ff MSU Physics 231 Fall

25 Velocity in 2D r ( x, y) v r t x y (, ) ( v x, v y) t t MSU Physics 231 Fall

26 MSU Physics 231 Fall d motion; decompose into horizontal and vertical components 1d motion ) ( t x x x x x at t v x at v v o o o t a t v x t a v v x ox x ox x t a t v y t a v v y oy y oy y

27 Near the surface of the earth with the y-axis pointing up a x = 0 a y = -g So for motion in x-direction v x x v ox v ox t and for motion in y direction the same equations (A-E) as before but using y (rather than x) MSU Physics 231 Fall

28 (A) v y v 0 y at positive y (B) y v t oy 1 at 2 2 points up so a = g (C) y v t y 1 at 2 2 (D) y 1 2 ( v v ) t oy y (E) y 1 2a ( v 2 y v 2 0 y ) MSU Physics 231 Fall

29 Pop and Drop MSU Physics 231 Fall

30 A B Pop and Drop For A: v y = - ½gt 2 v x = 0 y = y 0 -½gt 2 x= 0 For B: v y = - ½gt 2 v x = v 0 y = y 0 ½gt 2 x=x 0 t MSU Physics 231 Fall

31 Clicker Quiz Galileo and Newton stand on top of the tower of Pisa. Galileo drops a stone of mass 2 kg straight down (no initial velocity). Newton throws a 2 kg stone with an initial horizontal velocity of 3 m/s. Which stone will hit the ground first? (ignore effects of friction) a) The stone thrown by Galileo b) The stone thrown by Newton c) Both stones arrive at the same time d) Not enough information to say. MSU Physics 231 Fall

32 Clicker Quiz Galileo and Newton stand on top of the tower of Pisa. Galileo drops a stone of mass 2 kg straight down (no initial velocity). Newton throws a 2 kg stone with an initial horizontal velocity of 3 m/s. Which stone will hit the ground first? (ignore effects of friction) a) The stone thrown by Galileo b) The stone thrown by Newton c) Both stones arrive at the same time d) Not enough information to say. MSU Physics 231 Fall

33 Clicker Quiz A cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball? Firing Balls I a) it depends on how fast the cart is moving b) it falls behind the cart c) it falls in front of the cart d) it falls right back into the cart MSU Physics 231 Fall

34 Clicker Quiz A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball? Firing Balls I a) it depends on how fast the cart is moving b) it falls behind the cart c) it falls in front of the cart d) it falls right back into the cart In the frame of reference of the cart, the ball only has a vertical component of velocity. So it goes up and comes back down. To a ground observer, both the cart and the ball have the same horizontal velocity, so the ball still returns into the cart. when viewed from cart when viewed from ground MSU Physics 231 Fall

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36 Relative motion Motion is relative to a reference frame! Motion of the ball in rest-frame of cart Resulting motion Motion of the cart MSU Physics 231 Fall

37 When studying motion in 2D it is often convenient to decomposition the motion into horizontal and vertical components, both of which are both described in 1D Remember that the object can accelerate in one direction, but remain at the same speed in the other direction. Remember that after decomposition of 2D motion into horizontal and vertical components, you should investigate both components to understand the complete motion of a particle. After decomposition into horizontal and vertical directions, treat the two directions independently but the time is common to both. MSU Physics 231 Fall

38 Parabolic motion MSU Physics 231 Fall

39 Parabolic motion v = (v x, v y ) v x =v 0 cos v y =v 0 sin v x =v 0 cos v y =v 0 sin-1g v x =v 0 cos v y =v 0 sin-2g=0 v x =v 0 cos v y =v 0 sin-3g v x (t) = v 0x v y (t) = v 0y g t V x remains constant throughout the flight! v x =v 0 cos v y =v 0 sin-4g t=0 t=1 t=2 t=3 t=4 MSU Physics 231 Fall

40 Extra Credit Quiz You throw a ball in an arch ( up is the positive y direction), at the highest point in the arc the y component of the acceleration is: a) Impossible to tell b) -9.8 m/s 2 c) 9.8 m/s 2 d) 0 MSU Physics 231 Fall

41 Parabolic motion Where is the speed 1) highest? t=0 t=1 t=2 t=3 E A B C D t=5 MSU Physics 231 Fall

42 Parabolic motion Where is the speed 1) lowest? t=0 t=1 t=2 t=3 E A B C D t=5 MSU Physics 231 Fall

43 Question A hunter aims at a bird that is some distance away and flying very high (i.e. consider the vertical position of the hunter to be 0), but he misses. If the bullet leaves the gun with a speed of v 0 and friction by air is negligible, with what speed v f does the bullet hit the ground after completing its parabolic path? v 0 v f MSU Physics 231 Fall

44 Answer First consider the horizontal direction: v 0x = v 0 cos() Since there is no friction, there is no change in the horizontal component: v x (t f ) = v 0x Next the vertical direction with equations A and B: v y (t) = v 0y gt y(t) = v oy t ½ gt 2 (g = 9.8 m/s 2 ) When the bullet hits the ground y(t f )=0: 0 = v oy t f ½gt f 2 t f = 0 or t f = 2v 0y /g So, v y (t f ) = v 0y g(2v 0y /g) = v 0y Total speed = v 0x 2 + ( v 0y ) 2 = v 0!!!! The speed of the bullet has not changed, but the vertical component of the velocity has changed sign. MSU Physics 231 Fall

45 Clicker Quiz You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will: Dropping a Package a) quickly lag behind the plane while falling b) remain vertically under the plane while falling c) move ahead of the plane while falling d) not fall at all MSU Physics 231 Fall

46 What does the motion of the object look like according to the pilot? MSU Physics 231 Fall

47 Clicker Quiz You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will: Dropping a Package a) quickly lag behind the plane while falling b) remain vertically under the plane while falling c) move ahead of the plane while falling d) not fall at all Both the plane and the package have the same horizontal velocity at the moment of release. They will maintain this velocity in the x-direction, so they stay aligned. MSU Physics 231 Fall

48 Another example A football player throws a ball with initial velocity of 30 m/s at an angle of 30 o degrees w.r.t. the ground. 1) How far will the ball fly before hitting the ground? 2) What about an angle of 60 o? 3) At which angle is the distance thrown maximum? (A) (B) (C) x(t) = v 0 cos()t y(t) = v 0 sin()t ½gt 2 y(t) = 0 if t( v 0 sin() - ½gt )=0 t f =0 or t f = 2 v 0 sin()/g x(t f ) = v 0 cos() t f = 2 (v 0 ) 2 cos() sin() / g (D) if = 30 o x = 79.5 m if = 60 o x = 79.5 m!! (E) Maximum if cos() sin() is maximum, so =45 o x(=45 o ) = 91.7 m MSU Physics 231 Fall

49 And another example MSU Physics 231 Fall

50 And another example Vertical direction y(t) = y 0 + v 0 sin()t - ½gt 2 11 = sin(59)t - ½9.8t 2 4.9t t - 26=0 t = 3.35 (solve quadratic equation) This time correspond to how long it takes for the ball to land on the second building. MSU Physics 231 Fall

51 And another example Vertical direction y(t) = y 0 + v 0 sin()t - ½gt 2 11 = sin(59)t - ½9.8t 2 4.9t t - 26=0 t = 3.35 (solve quadratic equation) This time correspond to how long it takes for the ball to land on the second building. Calculate how far the ball goes in the horizontal direction. Horizontal direction x(t) = x 0 + v 0 cos() t use time derived from vertical direction x(3.35)= cos(59) 3.35 x(3.35)=17.4 m MSU Physics 231 Fall

52 Shoot the monkey The hunter aims his gun exactly at the monkey At the moment the hunter fires, the monkey drops off the branch. What happens? At the moment he fires, the monkey drops off the branch. What happens? a) monkey gets hit b) bullet goes over the monkey c) bullet goes under the monley d) no idea 52 PHY 231 MSU Physics 231 Fall

53 The horizontal position of the bullet is: x(t) = v 0 cos() t when x(t) = d then t d = d/[v 0 cos()] and its vertical position is: y(t) = v 0 sin() t ½gt 2 y(t d ) = d tan() ½gt 2 = h - ½gt 2 y=h v 0 X=0 X=d y=0 The vertical position of the monkey is: y(t) = h ½gt 2 The horizontal position is d 53 PHY 231 MSU Physics 231 Fall

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56 Uniform circular motion with constant speed Consider a car at constant speed, constrained to a circular track Velocity is always determined by the direction the car is facing at a given time Velocity is constantly changing, so this implies an acceleration But it s just the direction of the velocity that s changing! r MSU Physics 231 Fall

57 Uniform circular motion with constant speed v s r r arc length v c 2 r circumference T 2 r period v time for one turn MSU Physics 231 Fall

58 Angular Frequency Earth orbits the sun with a period of 365 days. T=365 days(24 h/day)(3600 s/h) = sec Frequency: f = 1/T = 1/( sec) = Hz R Earth travels at constant speed throughout its orbit: v = x/t It must traverse the circumference of the orbit: c = 2 π R Thus, the speed v = c/t = 2 π R / T We can also express this in terms of an angular frequency: The angular frequency = / t = 2 π / T = the speed at which the angle is changing Units = radians/s f= 1/T = 2 π f MSU Physics 231 Fall

59 Clicker Question A hummingbird with mass 4g flies northwest at a speed of 10 m/s and at an angle of 30 o relative to the horizontal. It beats its wings once every sec. Assuming it flies at constant velocity, what is the beat frequency of its wings? a) 0.14 Hz b) 7.1 Hz c) 14 Hz d) 71 Hz e) None of the above f= 1/T T = sec f = 1/0.014 sec Frequency: f = 71 Hz MSU Physics 231 Fall

60 MSU Physics 231 Fall Uniform circular motion with constant speed v r r v t t r t s v angular velocity arc length r s v

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63 Δ Δ Sin( /2 ) = ( s/2 ) / r Sin( /2 ) = ( v/2 ) / v MSU Physics 231 Fall

64 Δ Δ ( s/2 ) / r = sin( /2 ) sin( /2 ) = ( v/2 ) / v ( s/2 ) / r = ( v/2 ) / v ( s/t ) v/r = ( v/t ) ( s/t ) = v MSU Physics 231 Fall

65 Uniform circular motion with v constant speed D A a c B C Which way does the centripetal acceleration point? MSU Physics 231 Fall

66 MSU Physics 231 Fall radius 2 r T r v in radians with arclength r s t r t s v acceleration centripetal ) / ( 2 2 r r v a c angular velocity in (rad/s) 2 T r v t Uniform circular motion with constant speed period time for one turn T

67 Δ Δ sin( /2 ) = ( s/2 ) / r sin( /2 ) = ( v/2 ) / v MSU Physics 231 Fall

68 Δ Δ ( s/2 ) / r = sin( /2 ) sin( /2 ) = ( v/2 ) / v ( s/2 ) / r = ( v/2 ) / v ( s/t ) v/r = ( v/t ) ( s/t ) v MSU Physics 231 Fall

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70 Clicker Question begin Which route is shorter? a) Red b) Black c) The same d) Don t know end MSU Physics 231 Fall