Fundamentals of Physics School of Physical Science and Technology Mechanics (Bilingual Teaching) 张昆实 School of Physical Science and Technology Yangtze University

Chapter 4 Motion in Two and Three Dimensions 4-1 Moving in Two or Three Dimensions 4-2 Position and Displacement 4-3 Average Velocity and Instantaneous Velocity 4-4 Average Acceleration and Instantaneous Acceleration

Chapter 4 Motion in Two and Three Dimensions 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

4-1 Moving in Two or Three Dimensions ★ Objects move in two or three dimensions. ★The moving object is either a particle or an object that moves like a particle. ★To describ the motion of an object a reference frame must be chosen and a coordinate system must be constructed on it.

4-2 Position and Displacement ★ Position vector a vector extends from The origin of a coordinate system to the particle ● unit vector ● scalar components * (4-1) magnitude：

4-2 Position and Displacement ★ Position vector * magnitude： direction：

4-2 Position and Displacement 1 (4-2) (4-3) (4-4)

4-3 Average Velocity and Instantaneous Velocity If a particle moves through a displacement in a time Interval ,Then its ● Average displacement Velocity time interval (4-8) (4-9)

4-3 Average Velocity and Instantaneous Velocity ● (4-10) ● The direction of is always tangent to the particle’s path at the particle’s position (4-11)

4-3 Average Velocity and Instantaneous Velocity and its components (4-12) Fig.4-5 The velocity of a particle, along with the scalar components of

4-4 Average Acceleration and Instantaneous Acceleration When a particle’s velocity changes from to in a time interval ,Then its average change in velocity acceleration time interval ● t1 t2 (4-15)

4-4 Average Acceleration and Instantaneous Acceleration (4-16) (4-17) (4-18)

4.5-4.6 Projectile Motion ★ Projectile Motion : A particle moves in a vertical plane with some initial velocity and an angle with respect to the horizontal axis but its acceleration is always the free-fall acceleration , which is downward. (4-19) (4-20)

★ In projectile motion , the horizontal motion and the vertical motion are independent of each other. ★ the horizontal motion: ★ the vertical motion: (4-21)

4.5-4.6 Projectile Motion ★ the vertical motion: (4-22) (4-23) (4-24)

4.5-4.6 Projectile Motion Fig. 4-12 The projectile bullet always hits the falling coconut

4.5-4.6 Projectile Motion The Equation of the Path (4-21) (4-22) Let Solving Eq.4-21 for t and Substituting into Eq.4-22: (4-25) This is the equation of a parabola, so the path(trajectory) is parabolic.

The horizontal distance R is maximum for a launch angle of 4.5-4.6 Projectile Motion The Horizontal Range: the horizontal distance the Projectile has traveled when it returns to its initial (launch) height. (4-21) (4-22) Eliminating t between these two equations yields (4-26) The horizontal distance R is maximum for a launch angle of

In air: the horizontal range, the maximum 4.5-4.6 Projectile Motion The Effects of the Air (Air resistance): In vacuum: thepath(trajectory) is parabolic. In air: the horizontal range, the maximum height of the path are much less. Air resistance force: density of air, the cross section area of the projectile, mainly the velocity of the body. In Air In vacuum

4.7 Uniform Circular Motion ★ Uniform circular Motion : A particle travel around a circle or a circular arc at constant (uniform) speed. The velocity changes only in direction, there are still an acceleration– centripetal acceleration. (4-32) (4-33) Period: the time for going around a circle exactly once (circumference of the circle)

4.9 Relative Motion in Two Dimension y In three dimensions: Two observers are watching a moving particle P from the origins of frames A and B, while B moves at , The corresponding axes of frame A and B remain parallel position vecter: B to A : P to A : P to B : y v BA P v r r o BA o o o PB PA o o x o o Frame B o o o o o x r Frame A o BA r BA r PA (4-41) r r + r = PA PB BA r PB

4.9 Relative Motion in Two Dimension y (4-41) r PB PA BA = + y v BA P Take the time derivative + r = r r PA PB BA r r o o Get : o o PB PA o o x o o Frame B (4-42) o o v PA = BA + PB o o o x r Frame A Take the time derivative o BA a PA = PB (4-43) Since: v BA = constant, The acceleration of the particle measured from frames A and B are the same!

4.8 Relative Motion in One, Two Dimension in Two or Three Dimensions in One Dimension (4-38) x PB PA BA = + (4-41) r PB PA BA = + r BA PA PB = + + x x x = PA PB BA v PA = BA + PB (4-42) v PA = BA + PB (4-39) Since: v BA = constant, Since: v BA = constant, (4-40) a PA = PB (4-43) a a = PA PB The acceleration of the particle measured from frames A and B are the same!