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Physics 2048 Spring 2008. Lecture #4 Chapter 4 motion in 2D and 3D. Definitions Projectile motion Uniform circular motion Relative motion. Chapter 4 – 2D and 3D Motion. Definitions. Position vector: extends from the origin of a coordinate system to the particle.

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physics 2048 spring 2008
Physics 2048 Spring 2008

Lecture #4 Chapter 4 motion in 2D and 3D

chapter 4 2d and 3d motion

Definitions

  • Projectile motion
  • Uniform circular motion
  • Relative motion

Chapter 4 – 2D and 3D Motion

slide3

Definitions

Position vector: extends from the origin of a coordinate system to the particle.

Displacement vector: represents a particle’s position change during a certain

time interval.

Average velocity:

slide4

Instantaneous velocity:

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

Average acceleration:

Instantaneous acceleration:

slide5

Motion of a particle launched with initial velocity, v0 and free fall acceleration

g.

II. Projectile motion

The horizontal and vertical motions are independent from each other.

- Horizontal motion:ax=0  vx=v0x= cte

Range (R):horizontal distance traveled by a

projectile before returning to launch height.

- Vertical motion:ay= -g

slide6

- Trajectory: projectile’s path.

- Horizontal range: R = x-x0; y-y0=0.

(Maximum for a launch angle of 45º )

Overall assumption: the air through which the projectile moves has no effect

on its motion  friction neglected.

slide7

## In Galileo’s Two New Sciences, the author states that “for elevations (angles of projection) which exceed

or fall short of 45º by equal amounts, the ranges are equal…” Prove this statement.

y

v0

θ=45º

x

x=R=R’?

slide8

III. Uniform circular motion

Motion around a circle at constant speed.

Magnitude of velocity and acceleration constant.

Direction varies continuously.

-Velocity: tangent to circle in the direction of motion.

- Acceleration: centripetal

- Period of revolution:

v0y

v0x

slide9

1- A cat rides a merry-go-round while turning with uniform circular motion. At time t1= 2s, the cat’s velocity is: v1= (3m/s)i+(4m/s)j, measured on an horizontal xy coordinate system. At time t2=5s its velocity is:

v2= (-3m/s)i+(-4m/s)j. What are (a) the magnitude of the cat’s centripetal acceleration and (b) the cat’s average acceleration during the time interval t2-t1?

v2

In 3s the velocity is reversed  the cat reaches the opposite

side of the circle

x

v1

y

slide10

IV. Relative motion

Particle’s velocity depends on reference frame

1D

Frame moves at constant velocity

0

Observers on different frames of reference measure the same acceleration

for a moving particle if their relative velocity is constant.