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Physics 2048 Spring 2008

This lecture covers the definitions and concepts of motion in 2D and 3D, including projectile motion, uniform circular motion, and relative motion. Topics include position and displacement vectors, average and instantaneous velocity, acceleration, and the effects of air resistance. The Galilean principle of equal ranges for launch angles exceeding or falling short of 45º is proven. The concepts of uniform circular motion and relative motion are also explored, with examples and calculations.

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Physics 2048 Spring 2008

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  1. Physics 2048 Spring 2008 Lecture #4 Chapter 4 motion in 2D and 3D

  2. Definitions • Projectile motion • Uniform circular motion • Relative motion Chapter 4 – 2D and 3D Motion

  3. 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:

  4. 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:

  5. 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

  6. - 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.

  7. ## 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’?

  8. 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

  9. 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? v1 In 3s the velocity is reversed  the cat reaches the opposite side of the circle x v2 y

  10. 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.

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