Chapter 3 Kinematics in Two Dimension

1. Chapter 3 Kinematics in Two Dimension Ying Yi PhD PHYS I @ HCC

2. Outline • Review on Vectors (Adding, Subtracting,…) • Motion in two dimension • Displacement • Velocity • Acceleration • Projectile Motion • Relative Velocity PHYS I @ HCC

3. Adding Vectors Geometrically • Choose a scale • Draw the first vector with the appropriate length and direction, with respect to a coordinate system • Draw the next vector using tip-to-tail principle to • The resultant is drawn from the origin of to the end of the last vector PHYS I @ HCC

4. Vector Subtraction • Special case of vector addition • Add the negative of the subtracted vector • Continue with standard vector addition procedure PHYS I @ HCC

5. Multiplying or Dividing a Vector by a Scalar • The result of the multiplication or division is a vector • The magnitude of the vector is multiplied or divided by the scalar • If the scalar is positive, the direction of the result is the same as of the original vector • If the scalar is negative, the direction of the result is opposite that of the original vector PHYS I @ HCC

6. Components of a Vector • A component is a part • It is useful to use rectangular components • These are the projections of the vector along the x- and y-axes PHYS I @ HCC

7. More About Components, cont. • The components are the legs of the right triangle whose hypotenuse is • The value will be correct only if the angle lies in the first or fourth quadrant • In the second or third quadrant, add 180° PHYS I @ HCC

8. Motion in Two Dimensions • Using + or – signs is not always sufficient to fully describe motion in more than one dimension • Vectors can be used to more fully describe motion • Still interested in displacement, velocity, and acceleration PHYS I @ HCC

9. Displacement • The position of an object is described by its position vector, • The displacement of the object is defined as the change in its position • Units: m PHYS I @ HCC

10. Velocity • The average velocity is the ratio of the displacement to the time interval for the displacement • The instantaneous velocity is the limit of the average velocity as Δt approaches zero • The direction of the instantaneous velocity is along a line that is tangent to the path of the particle and in the direction of motion • Units: m/s PHYS I @ HCC

11. Acceleration • The average acceleration is defined as the rate at which the velocity changes • The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero • Units: m/s2 PHYS I @ HCC

12. Kinematics in Two Dimensions PHYS I @ HCC

13. Notes on two dimensional motion • It is important to realize that the x part of the motion occurs exactly as it would if the y part did not occur at all. Similarly, the y part of the motion occurs exactly as it would if the x part of the motion did not exist. • The independence of the x and y motions lies at the heart of two-dimensional kinematics. PHYS I @ HCC

14. Example 3.1 displacement In Figure 3.5, the dimensions to the right and upward are the positive directions. In the x direction, the spacecraft has an initial velocity component of v0x=+22 m/s and an acceleration component of ax=+24 m/s2. In the y direction, the analogous quantities are v0y=+14 m/s and ay=+12m/s2. At a time of t=7.0 s, find the x and y components of the spacecraft’s displacement. PHYS I @ HCC

15. Example 3.2 Velocity In Figure 3.5, In the x direction, the spacecraft has an initial velocity component of v0x=+22 m/s and an acceleration component of ax=+24 m/s2. In the y direction, the analogous quantities are v0y=+14 m/s and ay=+12m/s2. At a time of t=7.0 s, find the spacecraft’s final velocity (magnitude and direction). PHYS I @ HCC

16. Projectile Motion • An object may move in both the x and y directions simultaneously • It moves in two dimensions • The form of two dimensional motion we will deal with is an important special case called projectile motion PHYS I @ HCC

17. Assumptions of Projectile Motion • We may ignore air friction • We may ignore the rotation of the earth • With these assumptions, an object in projectile motion will follow a parabolic path PHYS I @ HCC

18. Rules of Projectile Motion • The x- and y-directions of motion are completely independent of each other • The x-direction is uniform motion • ax = 0 • The y-direction is free fall • ay = -g • The initial velocity can be broken down into its x- and y-components PHYS I @ HCC

19. Projectile Motion PHYS I @ HCC

20. Projectile Motion at Various Initial Angles • Complementary values of the initial angle result in the same range • The heights will be different • The maximum range occurs at a projection angle of 45o PHYS I @ HCC

21. Velocity of the Projectile • The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point • Remember to be careful about the angle’s quadrant PHYS I @ HCC

22. Projectile Motion Summary • Provided air resistance is negligible, the horizontal component of the velocity remains constant • Since ax = 0 • The vertical component of the velocity vy is equal to the free fall acceleration –g • Projectile motion can be described as a superposition of two independent motions in the x- and y-directions PHYS I @ HCC

23. Problem-Solving Strategy • Select a coordinate system and sketch the path of the projectile • Include initial and final positions, velocities, and accelerations • Resolve the initial velocity into x- and y-components • Treat the horizontal and vertical motions independently • Follow the techniques for solving problems with constant velocity to analyze the horizontal motion of the projectile • Follow the techniques for solving problems with constant acceleration to analyze the vertical motion of the projectile PHYS I @ HCC

24. Example 3.3 Falling Care Package Figure 3.7 shows an airplane moving horizontally with a constant velocity of +115 m/s at an altitude of 1050 m. The directions to the right and upward have been chosen as the positive directions. The plane release a care package that falls to the ground along a curved trajectory. Ignore air resistance, determine the time required for the package to hit the ground. PHYS I @ HCC

25. Example 3.4 Falling Care Package Figure 3.7 shows an airplane moving horizontally with a constant velocity of +115 m/s at an altitude of 1050 m. The directions to the right and upward have been chosen as the positive directions. The plane release a care package that falls to the ground along a curved trajectory. Ignore air resistance, find the magnitude and directional angle of the final velocity that the package has just before it strikes the ground. PHYS I @ HCC

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27. Group Problem: Projectile motion An Alaskan rescue plane drops a package of emergency rations to stranded hikers, as shown in Figure below. The plane is traveling horizontally at 40.0 m/s at a height of 1.00×102 m above the ground. (a) Where does the package strike the ground relative to the point at which it was released? (b) What are the horizontal and vertical components of the velocity of the package just before it hits the ground? (c) Find the angle of the impact. PHYS I @ HCC

28. Example 3.9 A home run A baseball player hits a home run, and the ball lands in the left field seats, 7.5 m above the point at which it was hit. It lands with a velocity of 36 m/s at an angle of 28° below the horizontal. The positive directions are upward and to the right in the drawing. Ignoring air resistance, find the magnitude and direction of the initial velocity with which the ball leaves the bat. PHYS I @ HCC

29. Group Question: Angry Bird Birds and pigs are in the same level in this scene. Angry birds leaves slingshot at a speed of 6.00m/s. At which angle should you shoot out those birds in order to hit the first and third pig? 3.67 m 3.18 m PHYS I @ HCC

30. Group Problem: Motion in 2D A ball is thrown upward from the top of a building at an angle of 30.0° to the horizontal and with an initial speed of 20.0 m/s, as in Figure. The point of release is 45.0 m above the ground. (a) How long does it take for the ball to hit the ground. (b)Find the ball’s speed at impact. (c) Find the horizontal range of the stone. Neglect air resistance. PHYS I @ HCC

31. Relative Velocity • Relative velocity is about relating the measurements of two different observers • It may be useful to use a moving frame of reference instead of a stationary one • It is important to specify the frame of reference, since the motion may be different in different frames of reference • There are no specific equations to learn to solve relative velocity problems PHYS I @ HCC

32. 1D relative velocity PHYS I @ HCC

33. 2D relative velocity PHYS I @ HCC

34. Example 3.11 Crossing a river The engine of a boat drives it across a river that is 1800 m wide. The velocity of the boat relative to the water is 4.0 m/s, directed perpendicular to the current, as in Figure 3.17. The velocity of the water relative to the shore is 2.0 m/s. (a) What is the velocity of the boat relative to the shore? (b) How long does it take for the boat to cross the river? PHYS I @ HCC

35. Problem-Solving Strategy: Relative Velocity • Label all the objects with a descriptive letter • Look for phrases such as “velocity of A relative to B” • Write the velocity variables with appropriate notation • If there is something not explicitly noted as being relative to something else, it is probably relative to the earth PHYS I @ HCC

36. Problem-Solving Strategy: Relative Velocity, cont • Take the velocities and put them into an equation • Keep the subscripts in an order analogous to the standard equation • Solve for the unknown(s) PHYS I @ HCC

37. Homework 3,10,13,17,20,25,43,47,53 PHYS I @ HCC