Chapter 3 Intro to Vectors Notes Section 1-3 Notes

Scalars & Vectors Scalar – a quantity that has magnitude but no direction  Units must be expressed  Ex: mass, volume, speed, time, distance, counting numbers Vector – a quantity that has magnitude and direction  Units & direction must be expressed  Ex: velocity, acceleration, force, displacement

Module Vector Addition – One Dimension If an object’s motion involves more than one vector, these can be added to find a net description of the motion A person walks 8 km East and then 6 km East. Displacement = 14 km East A person walks 8 km East and then 6 km West. Displacement = 2 km

Adding Vectors Graphically When vectors are added the answer is called the resultant To add vectors graphically, they must be drawn tail to tip Ex: A bus travels north 10.0km and west 10.0km. What is the resultant displacement? Displacement = 14.1km to the northwest 10.0km R

Ex: A bus travels north 10.0km and west 10.0km. What is the resultant displacement? D= 14.1km, 135° from the horizontal

Module Graphical Method of Vector Addition Tail to Tip Method

Adding Vectors Graphically – a little more complicated with graph paper A student walks 150m NE to his friend’s house. The student then walks another 800m NW to get to the bus stop. What is the displacement? 150m 800m R

Properties of Vectors Vectors can be moved parallel to themselves in a diagram to get them tail to tip Ex: A dog swims across a flowing river. End Start River velocity Dog velocity End River velocity

Vectors can be added in any order. Ex: A bus travels from city A to city B. OR

To subtract a vector, add its opposite. Mathematically this means change from positive to negative or vice versa. Graphically it means switch the direction. Ex: An airplane traveling at 50m/s is slowed by a wind of 5m/s. 50m/s -5m/s 45m/s F p = 50N F fr = 10N F net = 40N to the right

Vectors that are multiplied by scalars result in vectors Ex: A customer tells a cab driver to drive twice as fast 2 x 12m/s = 24m/s Scalar Vector

Practice Problems – use graph paper across from your Cornell notes 1) A boat moves at 0.80m/s across a river that is flowing at 1.5m/s. What is the resultant velocity of the boat? (solve graphically and algebraically) 2) A telephone pole support cable is in the way of some construction workers. In order for the work to proceed, the cable must be moved 2 meters closer to the pole. If the pole is 10 meters tall and the cable is currently fastened to the ground 8 meters from the pole, how much will the workers need to cut off from the cable when they move it?

#2 Solution: diagram the problem

Solution The original cable length is or 12.8m. By moving the cable 2 meters closer to the pole, we shorten the overall length of the cable to Or 11.7m. Therefore, 1.1m must be cut off the cable.

Ch 3 Homework #1 Questions: 1,5,8,9 on pg 67 Problems: 1,5,8,9 on pg. 68 Due October 26 Ch 3 Vector Quiz on Oct 26

Chapter 3-Resolving Vectors Section 5,6,7 Notes

Pythagorean Theorem to Find a Resultant (vectors are perpendicular) Remember: c 2 = a 2 + b 2 (c = hypotenuse) c is the resultant vector Use the tangent function to find the direction  Tanθ = opp/adjor θ = tan -1 (opp/adj)

Example: A runner runs 8.5km to the east and then runs 2.5km to the north. What is the displacement? 8.5km 2.5km R θ

Resolving Vectors into Components Components – the horizontal (x) or vertical (y) parts of a vector The x and y components of a vector make a right triangle with the vector Horizontal component Vertical component Vector

We can now describe motion in terms of the horizontal motion and vertical motion Ex: An airplane takes off at an angle of 25° and a speed of 185mi/h. It’s ground speed is 168mi/h. It climbs up at 78mi/h. 168mi/h 78mi/h 185mi/h 25°

Module Signs of Components

Properties of Right Triangles SOH CAH TOA sinθ = opp/hyp cosθ = adj/hyp tanθ = opp/adj θ hyp adj opp

SOH CAH TOA Use the cosine function to get the x component Use the sine function to get the y component 168mi/h 78mi/h 185mi/h 25° Hyp = 185mi/h Adj = 168mi/h 25° Opp = 78mi/h Note: coordinate system has been changed so all is positive

Formulas for Resolving Components V x = V(cosθ) V y = V(sinθ) V x = 185(cos25°) V y = 185(sin25°) V x = 168 V y = mi/h 25°

Steps for Adding Vectors that are not Perpendicular We cannot use the pythagorean theorem, so we must 1) After sketching, resolve each vector into its x and y components R θ θ V1V1 V2V2 V 1y V 1x V 2x V 2y θ θ θ1θ1 θ2θ2

2)Add the x components and add the y components 3)Use the Pythagorean theorem to find the magnitude of the resultant V R 2 = Vx 2 + Vy 2 4)Find the direction of the resultant θ = tan -1 (Vy/Vx) 5)Does my answer make sense? R θ θ

Practice Problems 3-9. A man in a rowboat is trying to cross a river that flows due west with a strong current. The man starts on the south bank and is trying to reach the north bank directly north from his starting point. The boat must a)head due north, b)head due west, c)head in a northwesterly direction, d)head in a northeasterly direction. Use a sketch to justify your answer. 7 th edition (3-10,2,3)

3-1. A rural mail carrier leaves the post office and drives 22.0km in a northerly direction to the next town. She then drives in a direction 60.0° south of east for 47.0km to another town. What is her displacement from the post office? 7 th edition (3-2,3)

Ch 3 Homework #2 Questions Problems Ch 3 Test on

Physics Chapter 3 Projectile Motion Section 3 Notes

Projectile Motion- The two dimensional motion of an object that is thrown or launched into the air.

Module A projectile is an object moving horizontally under the influence of Earth's gravity; its path is a parabola. waterslide (1min): Mythbusters (5min):

Concept Facts about Projectile Motion Projectile motion is free fall with horizontal velocity (Neglect air resistance) Consider motion only after release and before it hits Analyze the vertical and horizontal components separately a=0 in the horizontal, so v x = constant a y = -g = -9.8m/s/s Object projected horizontally will reach the ground at the same time as one dropped vertically

If the ball returns to the y = 0 point, then v = v 0. Range (x displacement) is determined by time it takes for projectile to return to ground or other end point

Formulas for Projectile Motion Quadratic Formula: ax 2 +bx+c = 0 x = -b± b 2 -4ac 2a Horizontal Motion Vertical Motion

Solving Projectile Motion Problems 1. Draw a picture and select a coordinate system. 2. Write down all the given information.

3. Resolve any vectors (velocity) into x and y components. 4. Choose the formula you need. 5. Solve the problem.

Ex: James thought it was the perfect murder plot: push his wife off a cliff and claim she fell. Ahh, but physics won, and James is in jail. James pushed his wife straight out with a velocity of 0.559m/s. She fell 182m. How far from the cliff did she land?

Practice Problems 3-5. A child sits upright in a wagon which is moving to the right at constant speed. The child extends her hand and throws an apple straight upward while the wagon continues to move forward at constant speed. If air resistance is neglected, will the apple land a) behind the wagon, b)in the wagon, or c) in front of the wagon A boy on a small hill aims his water-balloon slingshot horizontally, straight at a second boy hanging from a tree branch a distance d away. The boy in the tree is the same height up in the air as the water balloon sling shot on the hill. At the instant the water balloon is released, the second boy lets go and falls from the tree, hoping to avoid being hit. Use a diagram and 2-3 sentence explanation to explain that he made the wrong move. 7 th edition (3-6,7)

Practice Problems 3-3. A movie stunt driver on a motorcycle speeds horizontally off a 50.0m high cliff. How fast must the motorcycle leave the cliff-top if it is to land on level ground below, 90.0m from the base of the cliff where the cameras are A football is kicked at an angle θ=37.0° with a velocity of 20.0m/s. Calculate a) the maximum height, b) the time of travel before the football hits the ground, c) how far away it hits the ground, d)the velocity vector at its maximum height, and e) the acceleration vector at maximum height. Assume the ball leaves the foot at level ground and ignore air resistance and rotation of the ball. 7 th edition (3-4,5)

Practice Problems 3-8. Suppose the football in example 3-4 (θ=37.0° with a velocity of 20.0m/s) was a punt and left the punter’s foot at a height of 1.00m above the ground. How far did the football travel before hitting the ground? Set x 0 = 0, y 0 = 0. 7 th edition (3-9) Requires Quadratic

Old Practice Problems 1. People in movies often jump from buildings into pools. If a person jumps from the 10 th floor (30.0m) to a pool that is 5.0m from the building, with what initial speed must the person jump?

Practice #10 2. A golfer practices driving balls off a cliff into the water below. The cliff is 15m from the water. If the ball is launched at 51m/s at an angle of 15°, how far does the ball travel horizontally before hitting then water?

Practice #10 3. A missile is shot at a 60.0° angle and a speed of 55.0m/s. It is trying to reach an aircraft that is flying at 450m. Will the missile be able to reach its target? If not, what should be the initial velocity to reach its target?