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3. Motion in Two and Three Dimensions

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3. Motion in Two and Three Dimensions * R = Range Impact point Projectile Motion under Constant Acceleration * Projectile Motion under Constant Acceleration Strategy ... – PowerPoint PPT presentation

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Title: 3. Motion in Two and Three Dimensions


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 Earths surface.

5
Recap Example
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
Lets measure time from when the ball is
launched. This defines t 0.
Lets choose y 0 to be ground level and up to
be the positive y direction.
7
Recap Example
1. How high above the ground will the ball
reach?
with a g and v 0.
use
8
Recap Example
2. How long does it take the ball to reach
the ground?
Use
with a g and y 0.
9
Recap Example
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 .
13
Displacement
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
Acos? 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
Velocity of plane relative to air
Velocity of air relative to ground
Velocity of plane relative to ground
27
Example Relative Motion
A pilot wants to fly a plane due north Airspeed
200 km/h Windspeed 90 km/h direction W
to E 1. Flight heading? 2. Groundspeed?
Coordinate system î points from west to east
and j points from south to north.
28
Example Relative Motion
29
Example Relative Motion
30
Example Relative Motion
  • Equate x components
  • 0 200 sin ? 90
  • ? sin-1(90/200)
  • 26.7o west of north.

31
Example Relative Motion
Equate y components v 200 cos ? 179 km/h
32
Projectile Motion
33
Projectile Motion under Constant Acceleration
Coordinate system î points to the right, j
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
y(t)
y0
38
How to Shoot a Monkey
x 50 m h 10 m H 12 m Compute minimum initia
l velocity
H
39
Uniform Circular Motion
40
Uniform Circular Motion
O
r Radius
41
Uniform Circular Motion
Velocity
42
Uniform Circular Motion
d?/dt is called the angular velocity
43
Uniform Circular Motion
Acceleration
Assume d?/dt is constant
44
Uniform Circular Motion
Acceleration is towards center
Centripetal Acceleration
45
Uniform Circular Motion
The magnitudes of velocity and centripetal
acceleration are related as follows
46
Uniform Circular Motion
The magnitude of velocity and period T related
as follows
47
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.

48
Uniform Circular MotionAlternative Derivation
49
Uniform Circular Motion
O
Now, take the derivative
r constant radius
and define the unit vector at right angles
to
P
50
Uniform Circular Motion
We can write
where R90 represents a 90o rotation
O
Since
it follows that
and so
P
where b is a constant
51
Uniform Circular Motion
The velocity can be written as
O
It is easy to show that b v/r
Now, take the derivative again to get the
acceleration
P
52
Uniform Circular Motion
The acceleration can be written as
O
The acceleration is centripetal
P
using the fact that R90R90 1
53
Uniform Circular Motion
The magnitudes of velocity and centripetal
acceleration are related as follows
O
P
54
Uniform Circular Motion
The magnitude of velocity and period T related
as follows
O
P
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