chapter 2 two-dimensional motion and vectors. vectors, shmectors objectives 1. distinguish between a...
TRANSCRIPT
![Page 1: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/1.jpg)
Chapter 2
Two-Dimensional Motion and Vectors
![Page 2: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/2.jpg)
Vectors, Shmectors
Objectives1. Distinguish between a scalar and a vector.2. Add and subtract vectors by using the graphical method.3. Multiply and divide vectors by scalars.
![Page 3: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/3.jpg)
Vectors, Schmectors
● We can categorize quantities into scalars and vectors.
● A scalar is a quantity that has magnitude, but no direction. Examples include speed, volume, and moles.
● A vector is a physical quantity that has both direction and magnitude. Examples include velocity, acceleration, and displacement.
● We will denote vectors in this class by placing an arrow above the variable.
![Page 4: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/4.jpg)
Adding Vectors, Schmectors
● When adding vectors, make sure they have the same units and describe similar qualities.
● Do not add velocity and acceleration or velocity in km/h and velocity in m/s.
● Chapter 1 discussed vector addition and subtraction in one dimension when we calculated the displacement of an object that moved in the positive and negative directions.
● We can also add and subtract vectors graphically.
![Page 5: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/5.jpg)
Adding Vectors, Schmectors
● A resultant is a vector that represents the sum of two or more vectors.
● When adding vectors graphically, first draw the two vectors to scale with arrows at the ends denoting direction of motion.
● Next, draw a vector that connects the base of the first vector to the point of the second vector.
● The magnitude of the resultant can be found by measuring it and multiplying it by your scale factor.
![Page 6: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/6.jpg)
Adding Vectors, Schmectors
● The direction of the resultant can be determined by measuring the angle between the resultant and the first vector or between the resultant and a chosen reference line.
● This triangular method of vector addition is called the polygon method, the head-to-tail method, or the tip-to-tail method.
![Page 7: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/7.jpg)
Properties of Vectors, Schmectors
● Consider a situation in which two or more vectors act at the same point.
● A resultant vector can be found that has the same total effect as the combination of the individual vectors.
![Page 8: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/8.jpg)
Vectors, Schmectors Concept Check
● Imagine looking down from the second level of an airport at a toy car moving at 0.80 m/s across a walkway that moves at 1.5 m/s. How can you determine the car's resultant velocity?
![Page 9: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/9.jpg)
Okay, I get it now...
● The car's resultant velocity will be the combination of the two independent motions.
● We can draw a graphical representation of the two vectors with the car's velocity along the y-axis and the walkway's velocity along the x-axis.
● Next, we draw a vector connecting the tail of the car's velocity vector to the head of the walkway's velocity vector and measure its length and angle.
![Page 10: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/10.jpg)
Properties of Vectors, Schmectors
● Vectors can be added in any order and the sum will always be the same.
● When vectors are subtracted, you add the negative of the vector.
● The negative direction will always be left (or west) or down.
● The positive direction will always be right (or east) or up.
● Vectors can be multiplied by scalars, with a vector being the result.
![Page 11: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/11.jpg)
Vector, Schmector Operations
Objectives1. Identify appropriate coordinate systems for solving problems with vectors.2. Apply the Pythagorean Theorem and tangent function to calculate the magnitude and direction of a resultant vector.3. Resolve vectors into components using sine and cosine functions.4. Add vectors that are not perpendicular.
![Page 12: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/12.jpg)
Coordinates
● We use the standard x- and y-axes for diagramming the motion of an object in two dimensions.
● There are no rules for applying coordinate systems to situations involving vectors. As long as you are consistent within a situation, the final answer will be correct.
![Page 13: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/13.jpg)
Pythagorean Theorem and Vectors, Schmectors
● We can use the Pythagorean Theorem to determine the magnitude of the resultant in a situation where the two vectors being added form a perpendicular.
● c2 = a2 + b2
● Or (length of hypotenuse)2 = (length of one leg)2 + (length of other leg)2
![Page 14: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/14.jpg)
Off on a Tangent Again, Mrs. Beane?
● We can determine the direction of the resultant of two vectors that interact at a right angle by using the tangent function.
● tan Θ = opp● adj● tangent of angle = opposite leg● adjacent leg● Θ = tan-1(opp/adj)
![Page 15: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/15.jpg)
Concept Check
A plane travels from Houston, Texas to Washington, DC, which is 1540 km east and 1160 km north of Houston. What is the total displacement of the plane?
![Page 16: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/16.jpg)
Concept Check
X = 1540 kmy = 1160 km
d = √(1540km2 + 1160km2)d = 1930 kmΘ = tan-1(1160 km/1540 km)Θ = 37.0° north of east
![Page 17: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/17.jpg)
Concept Check
Emily passes a soccer ball 6.0 m directly across the field to Kara. Kara then kicks the ball 14.5 m directly down the field to Luisa. What is the ball's total displacement as it travels between Emily and Luisa?
![Page 18: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/18.jpg)
Concept Check
Δx=6.0 mΔy=-14.5 m
c2=a2+b2
d2=Δx2+Δy2
d=√(Δx2+Δy2)d=√{(6.0 m)2+(-14.5 m)2}d=16 mΘ=tan-1(opp/adj)Θ=tan-1(Δy/Δx)Θ=tan-1(-14.5m/6.0m)=-67.5°
![Page 19: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/19.jpg)
Resolving Vectors into Components
● The components of a vector are the projections of a vector along the axis of a coordinate system.
● The y component is parallel to the y-axis and the x component is parallel to the x-axis.
● Components of vectors are called projections in math class.
● You can break a vector down into its components to analyze its motion; this process is called resolving the vector.
![Page 20: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/20.jpg)
Resolving Vectors
● We use the sine and cosine functions to find the magnitude of the components of a vector.
● sin Θ = opp/hyp● cos Θ = adj/hyp
![Page 21: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/21.jpg)
Concept Check
An arrow is shot from a bow at an angle of 25° above the horizontal with an initial speed of 45 m/s. Find the horizontal and vertical components of the arrow's initial velocity.
![Page 22: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/22.jpg)
Concept Checkv=45 m/sΘ=25°vx=?
Vy=?
sin Θ =vy/v
cos Θ =vx/v
vy=v sin Θ
vx=v cos Θ
vy=(45 m/s) sin (25) vx=(45
m/s) cos (25)vy=19 m/s
vx=41 m/s
![Page 23: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/23.jpg)
Concept Check
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?
![Page 24: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/24.jpg)
Concept Check
95 km/h
![Page 25: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/25.jpg)
Concept Check
What is the magnitude of the vertical component of the velocity of the plane in the previous problem?
![Page 26: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/26.jpg)
Concept Check
44 km/h
![Page 27: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/27.jpg)
Concept Check
A truck drives up a hill with a 15° incline. If the truck has a constant speed of 22 m/s, what are the horizontal and vertical components of the truck's
velocity?
![Page 28: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/28.jpg)
Concept Check
21 m/s; 5.7 m/s
![Page 29: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/29.jpg)
Concept Check
What are the horizontal and vertical components of a cat's displacement when the cat has climbed 5m
directly up a tree?
![Page 30: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/30.jpg)
Concept Check
0 m; 5 m
![Page 31: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/31.jpg)
Vectors that are not Perpendicular
● When adding vectors that do not form a right triangle, you must resolve each of the vectors into its x and y components.
● The components along each axis are then added together.
● The resultants magnitude can then be found by using the Pythagorean Theorem and the direction can be found by using the tangent function.
![Page 32: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/32.jpg)
Concept Check
A plane flies 118 km at 15.0° south of east and then flies 118 km at 35.0° west of north. Find the magnitude and direction of the total displacement of the plane.
![Page 33: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/33.jpg)
Concept Check
d=81kmΘ=55°
![Page 34: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/34.jpg)
Concept Check
A football player runs directly down the field for 35 m before turning to the right at an angle of 25° from his original direction and running an additional 15 m before getting tackled. What is the magnitude and direction of the runners total displacement?
![Page 35: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/35.jpg)
49 m at 7.3° to the right of downfield
![Page 36: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/36.jpg)
Concept Check
A plane travels 2.5 km at an angle of 35° to the ground and changes direction and travels 5.2 km at
an angle of 22° to the ground. What is the magnitude and direction of the plane's total
displacement?
![Page 37: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/37.jpg)
Concept Check
7.5 km at 26° above the horizontal
![Page 38: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/38.jpg)
Projectile Motion
Objectives1. Recognize examples of projectile motion.2. Describe the path of a projectile as a parabola.3. Resolve vectors into their components and apply the kinematic equations to solve problems involving projectile motion.
![Page 39: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/39.jpg)
Vectors and Kinematic Equations
● We can resolve vectors into their components, apply the kinematic equations to each component, and recombine the components to determine the result.
● We will use components to simplify projectile motion The curved path that an object follows when
thrown,launched, or otherwise projected near the surface of Earth.
![Page 40: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/40.jpg)
Paths of Projectiles
● Projectiles move in a path that curves in a parabola.
● If an object has an initial horizontal velocity, there will be horizontal motion for the entire motion of the projectile.
● We will consider the horizontal velocity of an object to be constant. (neglect air resistance)
● We divide the motion of a projectile into vertical and horizontal components to examine them.
![Page 41: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/41.jpg)
Vertical Motion of a Projectile that Falls From Rest
● To analyze the vertical motion of a projectile, we use the following:
● vy,f
=ayΔt
● vy,f
2=2ayΔy
● Δy= ½ ay(Δt)2
![Page 42: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/42.jpg)
Horizontal Motion of a Projectile
● Remember that an object's horizontal velocity is considered to remain constant, so
● vx=v
x,i=constant
● Δx=vxΔt
![Page 43: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/43.jpg)
Projectile Motion
● Realize that projectile motion is free fall with an initial horizontal velocity.
● To find the velocity of a projectile at any point during its flight, find the vector that has the known components.
● Specifically, use the Pythagorean theorem to find the magnitude, and use the tangent function to find the direction of the velocity.
![Page 44: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/44.jpg)
Concept Check
A baseball rolls off a 0.70 m high desk and strikes the floor 0.25 m away from the base of the desk. How fast was the ball rolling?
![Page 45: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/45.jpg)
Concept Check
0.66 m/s
![Page 46: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/46.jpg)
Concept Check
A cat chases a mouse across a 1.0 m high table. The mouse steps out of the way, and the cat slides off the table and strikes the floor 2.2 m from the edge of the table. When the cat slid off the table, what was its
speed?
![Page 47: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/47.jpg)
Concept Check
4.9 m/s
![Page 48: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/48.jpg)
Concept Check
A pelican flying along a horizontal path drops a fish from a height of 5.4 m. The fish travels 8.0 m
horizontally before it hits the water. What is the pelican's speed?
![Page 49: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/49.jpg)
Concept Check
7.6 m/s
![Page 50: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/50.jpg)
Concept Check
If the pelican in item 3 was traveling at the same speed but was only 2.7 m above the water, how far would the fish travel horizontally before hitting the
water?
![Page 51: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/51.jpg)
Concept Check
5.6 m
![Page 52: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/52.jpg)
Projectiles Launched at an Angle
● When an object is launched an at angle to the horizontal, the projectile has an initial vertical and horizontal velocity.
● Given that the initial velocity vector makes an angle Θ with the horizontal, we can resolve the vector into its components such that
● vx=v
x,i=v
icosΘ=constant
● Δx=(vicosΘ)Δt
● vy,f
=visinΘ + a
yΔt
● vy,f
2=vi2(sinΘ)2 + 2a
yΔy
![Page 53: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/53.jpg)
Projectiles Launched at an Angle
● Δy = (visinΘ)Δt + ½ a
y(Δt)2
![Page 54: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/54.jpg)
Concept Check
In a scene in an action movie, a stuntman jumps from the top of one building to the top of another
building 4.0 m away. After a running start, he leaps at a velocity of 5.0 m/s at an angle of 15° with
respect to the flat roof. Will he make it to the other roof, which is 2.5 m shorter than the building he
jumps from?
![Page 55: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/55.jpg)
Concept Check
Yes, Δy = -2.3 m
![Page 56: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/56.jpg)
Concept Check
A golfer hits a golf ball at an angle of 25.0° to the ground. If the golf ball covers a horizontal distance
of 301.5 m, what is the ball's maximum height? (Hint: At the top of its flight, the ball's vertical
velocity component will be zero.)
![Page 57: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/57.jpg)
Concept Check
35.1 m
![Page 58: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/58.jpg)
Concept Check
A baseball is thrown at an angle of 25° relative to the ground at a speed of 23.0 m/s. If the ball was
caught 42.0 m from the thrower, how long was it in the air? How high did the ball travel before being
caught?
![Page 59: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/59.jpg)
Concept Check
2.0 s; 4.8 m
![Page 60: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/60.jpg)
Concept Check
Salmon often jump waterfalls to reach their breeding grounds. One salmon starts 2.00 m from a waterfall that is 0.55 m tall and jumps at an angle of 32.0°. What must be the salmon's minimum speed
to reach the waterfall?
![Page 61: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/61.jpg)
Concept Check
6.2 m/s
![Page 62: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/62.jpg)
Relative Motion
Objectives1. Describe situations in terms of frame of reference.2. Solve problems involving relative velocity.
![Page 63: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/63.jpg)
Frames of Reference
● When an object in motion is being observed by a person in motion, the observed velocity or displacement will be skewed.
● Depending on the location from which you observe an object's motion, the motion can appear different.
● When calculating the relative velocity of an object, use v
ac= v
ab+v
bc The subscripts denote “with respect to”
![Page 64: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/64.jpg)
Relative Velocity
Consider a situation in which a car traveling at 90 km/h is passing a car traveling at 80 km/h. What would be the velocity of the faster car with respect to the slower car?
![Page 65: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/65.jpg)
Relative Velocity
10 km/h
![Page 66: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/66.jpg)
Concept Check
A plane flies northeast at an airspeed of 563.0 km/h. (Airspeed is the speed of an aircraft relative to the air.) A 48.0 km/h wind is blowing to the southeast. What is the plane's velocity relative to the ground?
![Page 67: Chapter 2 Two-Dimensional Motion and Vectors. Vectors, Shmectors Objectives 1. Distinguish between a scalar and a vector. 2. Add and subtract vectors](https://reader036.vdocuments.us/reader036/viewer/2022081419/56649e165503460f94b01c27/html5/thumbnails/67.jpg)
Concept Check
565.0 km/h at 40.1° north of east.