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

2. 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

3. 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. 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.

5. 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：

6. 4-2 Position and Displacement
★ Position vector *
magnitude：
direction：

7. 4-2 Position and Displacement1
(4-2)
(4-3)
(4-4)

8. 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)

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)

10. 4-3 Average Velocity and Instantaneous Velocity
and its components
(4-12)
Fig The velocity
of a particle, along with
the scalar components of

11. 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)

12. 4-4 Average Acceleration and Instantaneous Acceleration
(4-16)
(4-17)
(4-18)

13. 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)

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

15. Projectile Motion
★ the vertical motion:
(4-22)
(4-23)
(4-24)

16. Projectile Motion
Fig The projectile bullet always hits the falling coconut

17. 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.

18. The horizontal distance R is maximum for a launch angle of
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

19. In air: the horizontal range, the maximum
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

20. 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)

21. 4.9 Relative Motion in Two Dimensiony
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

22. 4.9 Relative Motion in Two Dimensiony
(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!

23. 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!