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Chapter 3. Motion in Two or Three Dimensions. Modified by Mike Brotherton. Goals for Chapter 3. To use vectors to represent the position of a body To determine the velocity vector using the path of a body To investigate the acceleration vector of a body

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Chapter 3

Chapter 3

Motion in Two or Three Dimensions

Modified by

Mike Brotherton


Goals for chapter 3
Goals for Chapter 3

  • To use vectors to represent the position of a body

  • To determine the velocity vector using the path of a body

  • To investigate the acceleration vector of a body

  • To describe the curved path of projectile

  • (To investigate circular motion – left for later!)

  • To describe the velocity of a body as seen from different frames of reference


Introduction
Introduction

  • What determines where a batted baseball lands?

  • If a cyclist is going around a curve at constant speed, is he accelerating?

  • How is the motion of a particle described by different moving observers?

  • We need to extend our description of motion to two and three dimensions.


Position vector
Position vector

  • The position vector from the origin to point P has components x, y, and z.


Average velocity figure 3 2
Average velocity—Figure 3.2

  • The average velocity between two points is the displacement divided by the time interval between the two points, and it has the same direction as the displacement.


Instantaneous velocity
Instantaneous velocity

  • The instantaneous velocity is the instantaneous rate of change of position vector with respect to time.

  • The components of the instantaneous velocity are vx = dx/dt, vy = dy/dt, and vz = dz/dt.

  • The instantaneous velocity of a particle is always tangent to its path.


Calculating average and instantaneous velocity
Calculating average and instantaneous velocity

  • A rover vehicle moves on the surface of Mars.

  • Follow Example 3.1.


Average acceleration
Average acceleration

  • The average acceleration during a time interval t is defined as the velocity change during t divided by t.


Instantaneous acceleration
Instantaneous acceleration

  • The instantaneous acceleration is the instantaneous rate of change of the velocity with respect to time.

  • Any particle following a curved path is accelerating, even if it has constant speed.

  • The components of the instantaneous acceleration are ax = dvx/dt, ay = dvy/dt, and az = dvz/dt.


Calculating average and instantaneous acceleration
Calculating average and instantaneous acceleration

  • Return to the Mars rover.

  • Can follow Example 3.2 in text in a similar way as before.

  • Do the acceleration vectors make sense?


Parallel and perpendicular components of acceleration
Parallel and perpendicular components of acceleration

  • Return again to the Mars rover.

  • Follow Example 3.3.


Projectile motion figure 3 15
Projectile motion—Figure 3.15

  • A projectile is any body given an initial velocity that then follows a path determined by the effects of gravity and air resistance.

  • Begin by neglecting resistance and the curvature and rotation of the earth.


The x and y motion are separable figure 3 16
The x and y motion are separable—Figure 3.16

  • The red ball is dropped at the same time that the yellow ball is fired horizontally.

  • The strobe marks equal time intervals.

  • We can analyze projectile motion as horizontal motion with constant velocity and vertical motion with constant acceleration: ax = 0and ay = g.


The equations for projectile motion
The equations for projectile motion

  • If we set x0 = y0 = 0, the equations describing projectile motion are shown at the right.

  • The trajectory is a parabola.


The effects of air resistance figure 3 20
The effects of air resistance—Figure 3.20

  • Calculations become more complicated.

  • Acceleration is not constant.

  • Effects can be very large.

  • Maximum height and range decrease.

  • Trajectory is no longer a parabola.


Tranquilizing a falling monkey
Tranquilizing a falling monkey

  • Where should the zookeeper aim?

  • Follow Example 3.10.


A body projected horizontally
A body projected horizontally

  • A motorcycle leaves a cliff horizontally.

  • Follow Example 3.6.


Height and range of a projectile
Height and range of a projectile

  • A baseball is batted at an angle.

  • Follow Example 3.7.


Maximum height and range of a projectile
Maximum height and range of a projectile

  • What initial angle will give the maximum height and the maximum range of a projectile?

  • Follow Example 3.8.

CAUTION, only true for flat surfaces!


Different initial and final heights
Different initial and final heights

  • The final position is below the initial position.

  • Follow Example 3.9 using Figure 3.25.


Relative velocity
Relative velocity

  • The velocity of a moving body seen by a particular observer is called the velocity relative to that observer, or simply the relative velocity.

  • A frame of reference is a coordinate system plus a time scale.


Relative velocity in one dimension
Relative velocity in one dimension

  • If point P is moving relative to reference frame A, we denote the velocity of P relative to frame A as vP/A.

  • If P is moving relative to frame B and frame B is moving relative to frame A, then the x-velocity of P relative to frame A is vP/A-x = vP/B-x + vB/A-x.


Relative velocity in two or three dimensions
Relative velocity in two or three dimensions

  • We extend relative velocity to two or three dimensions by using vector addition to combine velocities.

  • In Figure 3.34, a passenger’s motion is viewed in the frame of the train and the cyclist.


Flying in a crosswind
Flying in a crosswind

  • A crosswind affects the motion of an airplane.

  • Follow Examples 3.14 and 3.15.

  • Refer to Figures 3.35 and 3.36.


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