Your SlideShare is downloading. ×
  • Like
Chapter 3
Upcoming SlideShare
Loading in...5

Thanks for flagging this SlideShare!

Oops! An error has occurred.


Now you can save presentations on your phone or tablet

Available for both IPhone and Android

Text the download link to your phone

Standard text messaging rates apply


Published in Technology , Business
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
No Downloads


Total Views
On SlideShare
From Embeds
Number of Embeds



Embeds 0

No embeds

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

    No notes for slide


  • 1. Two-Dimensional Motion and Vectors
    Chapter 3 pg. 81-105
  • 2. What do you think?
    How are measurements such as mass and volume different from measurements such as velocity and acceleration?
    How can you add two velocities that are in different directions?
  • 3. Introduction to Vectors
    Scalar - a quantity that has magnitude but no direction
    Examples: volume, mass, temperature, speed
    Vector - a quantity that has both magnitude and direction
    Examples: acceleration, velocity, displacement, force
  • 4. Vector Properties
    Vectors are generally drawn as arrows.
    Length represents the magnitude
    Arrow shows the direction
    Resultant - the sum of two or more vectors
    Make sure when adding vectors that
    You use the same unit
    Describing similar quantities
  • 5. Finding the Resultant Graphically
    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.
    The resultant for the addition of a + b is shown to the left as c.
  • 6. Vector Addition
    Vectors can be moved parallel to themselves without changing the resultant.
    the red arrow represents the resultant of the two vectors
  • 7. Vector Addition
    Vectors can be added in any order.
    The resultant (d) is the same in each case
    Subtraction is simply the addition of the opposite vector.
  • 8. Sample Resultant Calculation
    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.
  • 9. 3.2 Vector Operations
  • 10. What do you think?
    What is one disadvantage of adding vectors by the graphical method?
    Is there an easier way to add vectors?
  • 11. Vector Operations
    Use a traditional x-y coordinate system as shown below on the right.
    The Pythagorean theorem and tangent function can be used to add vectors.
    More accurate and less time-consuming than the graphical method
  • 12. Pythagorean Theorem and Tangent Function
  • 13. Pythagorean Theorem and Tangent Function
    We can use the inverse of the tangent function to find the angle.
    θ= tan-1 (opp/adj)
    Another way to look at our triangle
    d2 =Δx2 + Δy2
  • 14. Example
    An archaeologist climbs the great pyramid in Giza. The pyramid height is 136 m and width is 2.30 X 102m. What is the magnitude and direction of displacement of the archaeologist after she climbs from the bottom to the top?
  • 15. Example
    Δy= 136m
    width is 2.30 X 102m for whole pyramid
    So, Δx = 115m
    d = ?? θ= ??
  • 16. Example
    d2 =Δx2 + Δy2
    d = √Δx2 + Δy2
    d = √ (115)2 +(136)2
    d = 178m
    θ= tan-1 (opp/adj)
    θ= tan-1 (136/115)
    θ= 49.78°
  • 17. Example
    While following the directions on a treasure map a pirate walks 45m north then turns and walks 7.5m east. What single straight line displacement could the pirate have taken to reach the treasure?
  • 18. Resolving Vectors Into Components
  • 19. Resolving Vectors into Components
    Component: the horizontal x and vertical yparts that add up to give the actual displacement
    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 that the angle is 35.0˚.
  • 20. Example
    Given: v= 95 km/h θ= 35.0°
    Unknown vx=??vy= ??
    Rearrange the equations
    sin θ= opp/ hyp or sin θ=vy/v
    opp=(sin θ) (hyp) or vy=(sin θ)(v)
    cosθ= adj/ hyp or cosθ= vx/v
    adj= (cosθ)(hyp) or vx= (cosθ)(v)
  • 21. Example
    vy=(sin θ)(v)
    vy= (sin35°)(95)
    vy= 54.49 km/h
    vx= (cosθ)(v)
    vx = (cos 35°)(95)
    vx = 77.82 km/h
  • 22. Example
    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?
  • 23. 3.3 Projectile Motion
  • 24. What do you think?
    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.
    Which coin will strike the floor first?
    Explain your reasoning.
    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?
  • 25. Projectile Motion
    Projectiles: objects that are launched into the air
    tennis balls, arrows, baseballs, javelin
    Gravity affects the motion
    Projectile motion:
    The curved path that an object follows when thrown, launched or otherwise projected near the surface of the earth
  • 26. Projectile Motion
    Path is parabolic if air resistance is ignored
    Path is shortened under the effects of air resistance
  • 27. Components of Projectile Motion
    As the runner launches herself (vi), she is moving in the x and y directions.
  • 28. Projectile Motion
    Projectile motion is free fall with an initial horizontal speed.
    Vertical and horizontal motion are independent of each other.
    Vertically the acceleration is constant (10 m/s2 )
    We use the 4 acceleration equations
    Horizontally the velocity is constant
    We use the constant velocity equations
  • 29. Projectile Motion
    Components are used to solve for vertical and horizontal quantities.
    Time is the same for both vertical and horizontal motion.
    Velocity at the peak is purely horizontal (vy= 0).
  • 30. Example
    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?
  • 31. Example
    Given: d = 321m a = 10m/s2
    vi= 5m/s t = ?? vf = ??
    REMEMBER we need to figure out :
    Up and down aka free fall (use our 4 acceleration equations)
    Horizontal (use our constant velocity equation)
  • 32. Classroom Practice Problem (Horizontal Launch)
    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?
    Answer: 2.0 m/s
  • 33. Projectiles Launched at an Angle
    We will make a triangle and use our sin, cos, tan equations to find our answers
    Vy = V sin θ
    Vx = V cosθ
    tan = θ(y/x)
  • 34. Classroom Practice Problem(Projectile Launched at an Angle)
    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?
    Answer: 1.7 x 102m (170 m)