Motion in a Plane

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Chapter 3

• Motion in a Plane

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Motion in a Plane

• Vector Addition
• Velocity
• Acceleration
• Projectile motion

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Graphical Addition and Subtraction of Vectors
A vector is a quantity that has both a magnitude
and a direction. Position is an example of a
vector quantity.
A scalar is a quantity with no direction. The
mass of an object is an example of a scalar
quantity.
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Notation
Vector
The magnitude of a vector
The direction of vector might be 35? south of
east 20? above the x-axis or.
Scalar m (not bold face no arrow)
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Graphical Addition of Vectors
To add vectors graphically they must be placed
tip to tail. The result (F1 F2) points from
the tail of the first vector to the tip of the
second vector. This is sometimes called the
resultant vector R
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Vector Simulation
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Examples

• Trig Table
• Vector Components
• Unit Vectors

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Graphical Subtraction of Vectors
Think of vector subtraction A ? B as A(?B),
where the vector ?B has the same magnitude as B
but points in the opposite direction.
Vectors may be moved any way you please (to place
them tip to tail) provided that you do not change
their length nor rotate them.
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Velocity
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A particle moves along the blue path as shown.
At time t1 its position is ri and at time t2 its
position is rf.
y
x
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A displacement over an interval of time is a
velocity
The instantaneous velocity is represented by the
slope of a line tangent to the curve on the graph
of an objects position versus time.
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Acceleration
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A particle moves along the blue path as shown.
At time t1 its position is r0 and at time t2 its
position is rf.
y
vi
x
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A nonzero acceleration changes an objects state
of motion
These have interpretations similar to vav and v.
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Motion in a Plane with Constant Acceleration -
Projectile
What is the motion of a struck baseball? Once it
leaves the bat (if air resistance is negligible)
only the force of gravity acts on the
baseball. Acceleration due to gravity has a
constant value near the surface of the earth. We
call it g 9.8 m/s2 Only the vertical motion is
affected by gravity
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Projectile Motion
The baseball has ax 0 and ay ?g, it moves
with constant velocity along the x-axis and with
a changing velocity along the y-axis.
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Example An object is projected from the origin.
The initial velocity components are vix 7.07
m/s, and viy 7.07 m/s. Determine the x and y
position of the object at 0.2 second intervals
for 1.4 seconds. Also plot the results.
Since the object starts from the origin, ?y and
?x will represent the location of the object at
time ?t.
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Example continued
t (sec) x (meters) y (meters)
0 0 0
0.2 1.41 1.22
0.4 2.83 2.04
0.6 4.24 2.48
0.8 5.66 2.52
1.0 7.07 2.17
1.2 8.48 1.43
1.4 9.89 0.29
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Example continued
This is a plot of the x position (black points)
and y position (red points) of the object as a
function of time.
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Example continued
This is a plot of the y position versus x
position for the object (its trajectory). The
objects path is a parabola.
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Example (text problem 3.50) An arrow is shot
into the air with ? 60 and vi 20.0 m/s.
(a) What are vx and vy of the arrow when t 3
sec?
The components of the initial velocity are
CONSTANT
At t 3 sec
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Example continued
(b) What are the x and y components of the
displacement of the arrow during the 3.0 sec
interval?
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Example How far does the arrow in the previous
example land from where it is released?
The arrow lands when ?y 0.
Solving for ?t
What about the 2nd solution?
The distance traveled is
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Summary

• Adding and subtracting vectors (graphical method
  component method)
• Velocity
• Acceleration
• Projectile motion (here ax 0 and ay ?g)

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Projectiles Examples

• Problem solving strategy
• Symmetry of the motion
• Contact forces versus long-range forces
• Dropped from a plane
• The home run