Freely Falling Objects
Free fall from
rest:
Velocity increases linearly
Free fall from
x=0:
Maximum height as a function of initial velocity
If we throw a ball upward and recorded every position and associated time, we would produce the
resulting position versus time plot. It can be predicted exactly based on the following equations
Solve for t when v=0
time required to reach maximum after the object is launched
At the maximum height v=0
Maximum height We can determine total time for the ball to return to the ground
Initial conditions T=total round-trip time
Exactly twice the time
required to reach
maximum after the
object is launched
time required
for a round-trip
time required to
reach maximum
after the object
is launched
The time for the projectile to reach maximum height from ground exactly equals the time from the maximum height to the ground. You throw an object upward and the return time (hang time) is T
Then ½ of T is the time required for the object to reach the ground from it’s highest point in the trajectory. Symmetry Let t=0 at the highest point
Then v0=0 and x0=maximum
height h in the below equation
When the object hits the ground x=0, t = T/2
T=total round‐trip time
t0 fall time
Clicker Quiz
You notice droppings fall from Pigeons on the top of a tall
building. The time it takes for the droppings to hit the ground
is 3.0 seconds. Ignoring air resistance, how tall is the
building? Let g~10m/sec^2
A) 80meters
B) 20meters
C) 45meters
D) 11meters
E) none of the above
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You throw a ball upward at 25 m/s. Ignoring air
resistance what is the velocity after 2sec?
g
x
v0
t
What is the velocity after 3 sec?
Why is the velocity negative?
Scalar: number with units
Vector: quantity with magnitude and direction
How do we describe it’s motion?
You might say, “it went about a meter.”
A
But that is incomplete..
You should also say, it was traveling in a diagonal
line, straight path at an angle of 45 degrees.
The motion has a length and direction.
The mathematical name for such a quantity that has
both direction and magnitude or length is a vector.
The fly can be characterize by the width, height, how
much it weighs or the mass. These are scalars.
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Standard Vector Notation

A
•Text uses bold with arrow to denote a vector: •Also used for printing is simple bold print: A
•When dealing with just the magnitude of a vector in print, an italic letter will be 
used: A or | | A
– The magnitude of the vector has physical units.
– The magnitude is always a positive number.

A
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How to get to the library: need to know how far
and which way
r = displacement vector
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The Components of a Vector
Even though you know how far and in which
direction the library is, you may not be able
to walk there in a straight line:
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Information characterizing a vector is sometimes given in polar coordinates (r,θ)
If we want the vector described in Cartesian coordinates,
a transformation is needed in the form of trigonometric function.
Relate the angles of a triangle to the lengths of the sides of a triangle.
r
y
The most familiar trigonometric functions
are the sine, cosine, and tangent.
The sine function takes an angle and tells
the length of the y-component (rise) of
that triangle.
x
The cosine function takes an angle and
gives the length of x-component (run) of a
triangle.
The tangent function takes an angle and
tells the slope (y-component divided by
the x-component).
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Length, angle, and components can be calculated
from each other using trigonometry:
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The Components of a Vector
Resolve vector into perpendicular
components using a two-dimensional
Cartesian coordinate system:
rx=r cos(25)
ry=r sin (25)
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Clicker Quiz

Vector Ais 10 m long and points at 30° above the positive x‐axis. Determine the Ay component. A) 5.8m
B) 5.0m
C) 8.7m
D) 2.5m
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