Chapter 4 Motion in Two and Three Dimensions

4.1. What is Physics?      4.2. Position and Displacement      4.3. Average Velocity and Instantaneous Velocity      4.4. Average Acceleration and Instantaneous

Acceleration      4.5. Projectile Motion      4.6. Projectile Motion Analyzed      4.7. Uniform Circular Motion      4.8. Relative Motion in One Dimension      4.9. Relative Motion in Two Dimensions

What is Physics?

Position and Displacement

Position vector:

Displacement :

EXAMPLE 1: Displacement

1r

2r

In Fig., the position vector for a particle is initially at

and then later is

What is the particle's displacement from

to ?

Problem 2

A rabbit runs across a parking lot on which a set of coordinate axes has, strangely enough, been drawn. The coordinates of the rabbit’s position as functions of time t (second) are given by

At t=15 s, what is the rabbit’s position vector in unit-vector notation and in magnitude-angle notation?

Average and Instantaneous Velocity

Instantaneous velocity is:

Particle’s Path vs Velocity

Displacement: The velocity vector

The direction of the instantaneous velocity of a particle is always tangent to the particle’s path at the particle’s position.

Problem 3

A rabbit runs across a parking lot on which a set of coordinate axes has, strangely enough, been drawn. The coordinates of the rabbit’s position as functions of time t (second) are given by

At t=15 s, what is the rabbit’s velocity vector in unit-vector notation and in magnitude-angle notation?

Average and Instantaneous Acceleration

Average acceleration is

Instantaneous acceleration is

yx zx y z

vv vva i j k a i a j a k

t t t t

                                                      

                                                      

Speed up or slow down

• If the velocity and acceleration components along a given axis have the same sign then they are in the same direction. In this case, the object will speed up.

• If the acceleration and velocity components have opposite signs, then they are in opposite directions. Under these conditions, the object will slow down.

Problem 4

A rabbit runs across a parking lot on which a set of coordinate axes has, strangely enough, been drawn. The coordinates of the rabbit’s position as functions of time t (second) are given by

At t=15 s, what is the rabbit’s acceleration vector in unit-vector notation and in magnitude-angle notation?

How to solve two-dimensional motion problem?

One ball is released from rest at the same instant that another ball is shot horizontally to the right

The horizontal and vertical motions (at right angles to each other) are independent, and the path of such a motion can be found by combining its horizontal and vertical position components.

By Galileo

Projectile Motion

A particle moves in a vertical plane with some initial velocity but its acceleration is always the free-fall acceleration g, which is downward. Such a particle is called a projectile and its motion is

called projectile motion.

Properties of Projectile Motion

The Horizontal Motion:•no acceleration •velocity vx remains unchanged from its initial value throughout the motion•The horizontal range R is maximum for a launch angle of 45°

The vertical Motion:•Constant acceleration g •velocity vy=0 at the highest point.

    

                                                                                                              

Check Your Understanding

A projectile is fired into the air, and it follows the parabolic path shown in the drawing. There is no air resistance. At any instant, the projectile has a velocity v and an acceleration a. Which one or more of the drawings could not represent the directions for v and a at any point on the trajectory?

Example 4  A Falling Care Package Figure 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 releases a “care package” that falls to the ground along a curved trajectory. Ignoring air resistance, (a). determine the time required for the package to hit the ground. (b) find the speed of package B and the direction of the velocity vector just before package B hits the ground.

Example 5  The Height of a Kickoff A placekicker kicks a football at an angle of θ=40.0o above the

horizontal axis, as Figure shows. The initial speed of the ball is

(a) Ignore air resistance and find the maximum height H that the ball attains.

(b) Determine the time of flight between kickoff and landing. (c). Calculate the range R of the projectile.

UNIFORM CIRCULAR MOTION

Uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path

                                                                                        

Properties of UNIFORM CIRCULAR MOTION

• Period of the motion T: is the time for a particle to go around a closed path exactly once has a special name.

Average speed is :

This number of revolutions in a given time is known as the frequency, f.

Example 6  A Tire-Balancing Machine

The wheel of a car has a radius of r=0.29 m and is being rotated at 830 revolutions per minute (rpm) on a tire-balancing machine. Determine the speed (in m/s) at which the outer edge of the wheel is moving.

CENTRIPETAL ACCELERATION • Magnitude: The centripetal acceleration

of an object moving with a speed v on a circular path of radius r has a magnitude ac given by 2

c

va

R

•Direction: The centripetal acceleration vector always points toward the center of the circle and continually changes direction as the object moves.

Check Your Understanding

The car in the drawing is moving clockwise around a circular section of road at a constant speed. What are the directions of its velocity and acceleration at following positions? Specify your responses as north, east, south, or west.

(a)position 1(b) position 2

Example 7  The Effect of Radius on Centripetal Acceleration

The bobsled track at the 1994 Olympics in Lillehammer, Norway, contained turns with radii of 33 m and 24 m, as Figure illustrates. Find the centripetal acceleration at each turn for a speed of 34 m/s, a speed that was achieved in the two-man event. Express the answers as multiples of g=9.8 m/s2.

Relative Motion in One Dimension

PA PB BAv v v

xPA xPB xBAa a a

                                                                                                             

The coordinate                   

The velocity                   

The acceleration                   

PA PB BAx x x

Relative Motion in Two Dimension

PA PB BAv v v

PA PB BAa a a

                                                                                                             

The coordinate                   

The velocity                   

The acceleration                   

PA PB BAr r r

Sample Problem In Fig. 4-23a, a plane moves due

east while the pilot points the plane somewhat south of east, toward a steady wind that blows to the northeast. The plane has velocity relative to the wind, with an airspeed (speed relative to the wind) of 215 km/h, directed at angle θ south of east. The wind has velocity vpG relative to the ground with speed of 65.0 km/h, directed 20.0° east of north. What is the magnitude of the velocity of the plane relative to the ground, and what is θ?