1. 3. Motion in Two and Three Dimensions

2. Recap: Constant Acceleration
Area under
the function
v(t).

3. Recap: Constant Acceleration

4. Recap: Acceleration due to Gravity (Free Fall)
In the absence of air resistance all objects fall with the same constant acceleration of about g = 9.8 m/s2
near the Earth’s surface.

5. Recap: Example
2 m
5 m/s
A ball is thrown upwards at 5 m/s, relative to the ground, from a height of 2 m.
We need to choose a coordinate system.

6. Recap: Example Let’s measure v0 = 5 m/s time from when
y0 = 2 m
v0 = 5 m/s y
Let’s measure
time from when
the ball is launched.
This defines t = 0.
Let’s choose y = 0 to be ground level
and up to be the positive y direction.

7. Recap: Example y0 = 2 m v0 = 5 m/s y
1. How high above the ground will the ball reach?
use
with a = –g
and v = 0.

8. Recap: Example y0 = 2 m v0 = 5 m/s y 2. How long does it
take the ball to
reach the ground?
Use
with a = –g
and y = 0.

9. Recap: Example y0 = 2 m v0 = 5 m/s y 3. At what speed
does the ball hit
the ground?
Use
with a = –g
and y = 0.

10. Vectors

11. Vectors A vector is a mathematical quantity that has two properties:
direction and magnitude.
One way to represent a vector
is as an arrow: the arrow gives
the direction and its length the
magnitude.

12. Position A position p is a vector: its direction is from o
to p and its length is the distance from o to p.
A vector is usually
represented by a
symbol like . p

13. Displacement
A displacement is another example of
a vector.

14. Vector Addition The order in which the vectors are added does
not matter, that is, vector
addition is commutative.

15. Vector Scalar Multiplication
a and –q are scalars (numbers).

16. Vector Subtraction If we multiply a vector by –1 we reverse its
direction, but keep its magnitude the same.
Vector subtraction
is really vector
addition with
one vector reversed.

17. Vector Components Acosq is the component, or the projection, of
the vector A along
the vector B.

18. Vector Components

19. Vector Addition using Components

20. Unit Vectors From the components, Ax, Ay, and Az, of a vector,
we can compute its length, A, using
If the vector A is multiplied
by the scalar 1/A we get a new
vector of unit length in the
same direction as vector A;
that is, we get a unit vector.

21. Unit Vectors It is convenient to define unit vectors
parallel to the x, y and z
axes, respectively.
Then, we can write a
vector A as follows:

22. Velocity and Acceleration Vectors

23. Velocity

24. Acceleration

25. Relative Motion

26. Relative Motion N S W E Velocity of plane relative to air
Velocity of air
relative to ground
Velocity of plane
relative to ground

27. Example – Relative MotionS W E
A pilot wants to fly plane due north
Airspeed: km/h
Windspeed: 90 km/h
direction: W to E
1. Flight heading?
2. Groundspeed?
Coordinate system: î points from
west to east and ĵ points from south to north.

28. Example – Relative MotionS W E

29. Example – Relative MotionS W E

30. Example – Relative MotionS W E
Equate x components
0 = –200 sinq + 90
q = sin-1(90/200)
= 26.7o west of north.

31. Example – Relative MotionS W E
Equate y components
v = 200 cosq
= 179 km/h

32. Projectile Motion

33. Projectile Motion under Constant Acceleration
Coordinate system:
î points to the right,
ĵ points upwards

34. Projectile Motion under Constant Acceleration
Impact point
R = Range

35. Projectile Motion under Constant Acceleration
Strategy: split motion into x and y components.
R = Range
R = x - x0
h = y - y0

36. Projectile Motion under Constant Acceleration
Find time of flight by solving y equation:
And find
range from:

37. Projectile Motion under Constant Acceleration
Special case: y = y0, i.e., h = 0 R y0
y(t)

38. Uniform Circular Motion

39. Uniform Circular Motion
r = Radius

40. Uniform Circular Motion
Velocity

41. Uniform Circular Motion
dq/dt is called the
angular
velocity

42. Uniform Circular Motion
Acceleration
For uniform motion
dq/dt is
constant

43. Uniform Circular Motion
Acceleration
is towards
center
Centripetal
Acceleration

44. Uniform Circular Motion
Magnitudes of velocity
and centripetal
acceleration are
related
as follows

45. Uniform Circular Motion
Magnitude of velocity
and period T related as
follows r

46. Summary
In general, acceleration changes both the magnitude and direction of the velocity.
Projectile motion results from the acceleration due to gravity.
In uniform circular motion, the acceleration is centripetal and has constant magnitude v2/r.

47. How to Shoot a Monkey H x = 50 m h = 10 m H = 12 m Compute minimum
initial
velocity H