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

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

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 3 PHYS HCC

4 Vector Subtraction Special case of vector addition Add the negative of the subtracted vector Continue with standard vector addition procedure

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 5 PHYS 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

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° 7 PHYS HCC

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 8 PHYS 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

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 Δ t 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 10 PHYS HCC

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 Δ t approaches zero Units: m/s 2 11 PHYS HCC

Kinematics in Two Dimensions PHYS HCC 12

Notes on two dimensional motion PHYS HCC 13 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.

Example 3.1 displacement PHYS HCC 14 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 v 0x =+22 m/s and an acceleration component of a x =+24 m/s 2. In the y direction, the analogous quantities are v 0y =+14 m/s and a y =+12m/s 2. At a time of t=7.0 s, find the x and y components of the spacecraft’s displacement.

Example 3.2 Velocity PHYS HCC 15 In Figure 3.5, In the x direction, the spacecraft has an initial velocity component of v 0x =+22 m/s and an acceleration component of a x =+24 m/s 2. In the y direction, the analogous quantities are v 0y =+14 m/s and a y =+12m/s 2. At a time of t=7.0 s, find the spacecraft’s final velocity (magnitude and direction).

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 16 PHYS HCC

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 17 PHYS HCC

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

Projectile Motion 19 PHYS 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 45 o

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 21 PHYS HCC

Projectile Motion Summary Provided air resistance is negligible, the horizontal component of the velocity remains constant Since a x = 0 The vertical component of the velocity v y 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 22 PHYS HCC

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 23 PHYS HCC

Example 3.3 Falling Care Package PHYS HCC 24 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.

Example 3.4 Falling Care Package PHYS HCC 25 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.

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Group Problem: Projectile motion PHYS HCC 27 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 1.00×10 2 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.

Example 3.9 A home run PHYS HCC 28 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 HCC 29 Group Question: Angry Bird 3.18 m 3.67 m 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?

Group Problem: Motion in 2D PHYS HCC 30 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.

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 31 PHYS HCC

1D relative velocity PHYS HCC 32

2D relative velocity PHYS HCC 33

Example 3.11 Crossing a river PHYS HCC 34 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 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?

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 35 PHYS HCC

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) 36 PHYS HCC

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