TwoDimensional Motion and Vectors Chapter 3 pg 81

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+ Two-Dimensional Motion and Vectors Chapter 3 pg. 81 -105

+ Two-Dimensional Motion and Vectors Chapter 3 pg. 81 -105

+ What do you think? n How are measurements such as mass and volume

+ What do you think? n How are measurements such as mass and volume different from measurements such as velocity and acceleration? n How can you add two velocities that are in different directions?

+ Introduction to Vectors n Scalar - a quantity that has magnitude but no

+ Introduction to Vectors n Scalar - a quantity that has magnitude but no direction n Examples: volume, mass, temperature, speed n Vector - a quantity that has both magnitude and direction n Examples: acceleration, velocity, displacement, force

+ Vector Properties n Vectors are generally drawn as arrows. n Length represents the

+ Vector Properties n Vectors are generally drawn as arrows. n Length represents the magnitude n Arrow shows the direction n Resultant - the sum of two or more vectors n Make n You sure when adding vectors that use the same unit n Describing similar quantities

+ Finding the Resultant Graphically n Method n n Draw each vector in the

+ Finding the Resultant Graphically n Method n n Draw each vector in the proper direction. Establish a scale (i. e. 1 cm = 2 m) and draw the vector the appropriate length. Draw the resultant from the tip of the first vector to the tail of the last vector. Measure the resultant. n The resultant for the addition of a + b is shown to the left as c.

+ Vector Addition n Vectors can be moved parallel to themselves without changing the

+ Vector Addition n Vectors can be moved parallel to themselves without changing the resultant. n the red arrow represents the resultant of the two vectors

+ Vector Addition n Vectors can be added in any order. n The resultant

+ Vector Addition n Vectors can be added in any order. n The resultant (d) is the same in each case n Subtraction is simply the addition of the opposite vector.

Sample Resultant Calculation n. A toy car moves with a velocity of. 80 m/s

Sample Resultant Calculation n. A toy car moves with a velocity of. 80 m/s across a moving walkway that travels at 1. 5 m/s. Find the resultant speed of the car.

+ 3. 2 Vector Operations

+ 3. 2 Vector Operations

+ What do you think? n What is one disadvantage of adding vectors by

+ What do you think? n What is one disadvantage of adding vectors by the graphical method? n Is there an easier way to add vectors?

+ Vector Operations n Use a traditional x-y coordinate system as shown below on

+ Vector Operations n Use a traditional x-y coordinate system as shown below on the right. n The Pythagorean theorem and tangent function can be used to add vectors. n More accurate and less time-consuming than the graphical method

+ Pythagorean Theorem and Tangent Function

+ Pythagorean Theorem and Tangent Function

+ Pythagorean Theorem and Tangent Function n We can use the inverse of the

+ Pythagorean Theorem and Tangent Function n We can use the inverse of the tangent function to find the angle. n θ= tan-1 (opp/adj) n Another n d 2 = Δx 2 way to look at our triangle + Δy 2 d θ Δx Δy

+ Example n An archaeologist climbs the great pyramid in Giza. The pyramid height

+ Example n An archaeologist climbs the great pyramid in Giza. The pyramid height is 136 m and width is 2. 30 X 102 m. What is the magnitude and direction of displacement of the archaeologist after she climbs from the bottom to the top?

+ Example n Given: n Δy= 136 m n width n So, is 2.

+ Example n Given: n Δy= 136 m n width n So, is 2. 30 X 102 m for whole pyramid Δx = 115 m n Unknown: nd = ? ? θ= ? ?

+ Example n Calculate: n θ= tan-1 (opp/adj) n d 2 =Δx 2 +

+ Example n Calculate: n θ= tan-1 (opp/adj) n d 2 =Δx 2 + Δy 2 n θ= tan-1 (136/115) nd = √Δx 2 + Δy 2 n θ= 49. 78° nd = √ (115)2 +(136)2 nd = 178 m

+ Example n While following the directions on a treasure map a pirate walks

+ Example n While following the directions on a treasure map a pirate walks 45 m north then turns and walks 7. 5 m east. What single straight line displacement could the pirate have taken to reach the treasure?

+ Resolving Vectors Into Components

+ Resolving Vectors Into Components

+ Resolving Vectors into Components n Component: the horizontal x and vertical yparts that

+ Resolving Vectors into Components n Component: the horizontal x and vertical yparts that add up to give the actual displacement n For the vector shown at right, find the vector components vx (velocity in the x direction) and vy (velocity in the y direction). Assume that the angle is 35. 0˚. 35°

+ Example n Given: v= 95 km/h n Unknown vx=? ? vy= ? ?

+ Example n Given: v= 95 km/h n Unknown vx=? ? vy= ? ? n Rearrange n sin the equations θ= opp/ hyp n opp=(sin n cosθ= n adj= θ) (hyp) adj/ hyp (cosθ)(hyp) θ= 35. 0°

+ Example nvy=(sin θ)(v) nvx= (cosθ)(v) n vy= (sin 35°)(95) n vx = (cos

+ Example nvy=(sin θ)(v) nvx= (cosθ)(v) n vy= (sin 35°)(95) n vx = (cos 35°)(95) n vy= 54. 49 km/h n vx = 77. 82 km/h

+ Example n How fast must a truck travel to stay beneath an airplane

+ Example n How fast must a truck travel to stay beneath an airplane that is moving 105 km/h at an angle of 25° to the ground?

+ 3. 3 Projectile Motion

+ 3. 3 Projectile Motion

+ What do you think? n Suppose two coins fall off of a table

+ What do you think? n Suppose two coins fall off of a table simultaneously. One coin falls straight downward. The other coin slides off the table horizontally and lands several meters from the base of the table. n n Which coin will strike the floor first? Explain your reasoning. n Would your answer change if the second coin was moving so fast that it landed 50 m from the base of the table? Why or why not?

+ Projectile Motion n Projectiles: n tennis balls, arrows, baseballs, javelin n Gravity affects

+ Projectile Motion n Projectiles: n tennis balls, arrows, baseballs, javelin n Gravity affects the motion n Projectile n objects that are launched into the air motion: The curved path that an object follows when thrown, launched or otherwise projected near the surface of the earth

+ Projectile Motion n Path is parabolic if air resistance is ignored n Path

+ Projectile Motion n Path is parabolic if air resistance is ignored n Path is shortened under the effects of air resistance

Components of Projectile Motion n As the runner launches herself (vi), she is moving

Components of Projectile Motion n As the runner launches herself (vi), she is moving in the x and y directions.

+ Projectile Motion n Projectile motion is free fall with an initial horizontal speed.

+ Projectile Motion n Projectile motion is free fall with an initial horizontal speed. n Vertical and horizontal motion are independent of each other. the acceleration is constant (-10 m/s 2 ) n We use the 4 acceleration equations n Horizontally the velocity is constant n We use the constant velocity equations n Vertically

+ Projectile Motion n Components are used to solve for vertical and horizontal quantities.

+ Projectile Motion n Components are used to solve for vertical and horizontal quantities. n Time is the same for both vertical and horizontal motion. n Velocity (vy= 0). at the peak is purely horizontal

+ Example n The Royal Gorge Bridge in Colorado rises 321 m above the

+ Example n The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas river. Suppose you kick a rock horizontally off the bridge at 5 m/s. How long would it take to hit the ground and what would it’s final velocity be?

+ Example n Given: d = 321 m vi= 5 m/s n REMEMBER n

+ Example n Given: d = 321 m vi= 5 m/s n REMEMBER n Up a = 10 m/s 2 t = ? ? vf = ? ? we need to figure out : and down aka free fall (use our 4 acceleration equations) n Horizontal (use our constant velocity equation)

+ Classroom Practice Problem (Horizontal Launch) n People in movies often jump from buildings

+ Classroom Practice Problem (Horizontal Launch) n People in movies often jump from buildings into pools. If a person jumps horizontally by running straight off a rooftop from a height of 30. 0 m to a pool that is 5. 0 m from the building, with what initial speed must the person jump? n Answer: 2. 0 m/s

+ Projectiles Launched at an Angle n We will make a triangle and use

+ Projectiles Launched at an Angle n We will make a triangle and use our sin, cos, tan equations to find our answers n Vy = V sin θ n Vx = V cosθ n tan = θ(y/x)

+ Classroom Practice Problem (Projectile Launched at an Angle) n. A golfer practices driving

+ Classroom Practice Problem (Projectile Launched at an Angle) n. A golfer practices driving balls off a cliff and into the water below. The edge of the cliff is 15 m above the water. If the golf ball is launched at 51 m/s at an angle of 15°, how far does the ball travel horizontally before hitting the water? n Answer: 1. 7 x 102 m (170 m)