Mathematical Analysis
of Motion
Physics 11
Uniform Motion
• This type of motion means that the velocity of an
object remains constant.
• We have so far only considered uniform motion
when solving mathematically problems.
• These problems consist of speed (velocity),
distance (displacement) and time problems.
Example 1
a) John travelled 42 km in 6 hours. How fast was he
going?
b) If John kept that speed for another 10 hours,
what extra distance would he travel?
c) If John increased his speed to 8 km/h, how long
would it take him to travel 72 km?
Example 2
An athlete runs a 100 m race in 10.3 s. Another
athlete is placed second with a time of 10.8 s.
Assuming that the athletes ran at their average
speed the whole race, find the distance separating
the two athletes when the 1st place runner arrives at
the finish line.
Non Uniform Motion
• This type of motion means that the velocity is
changing, either in magnitude or in direction.
• In this section, we will only discuss the change in
magnitude in velocity.
• In other words, we will for the most part neglect
direction, and thus refer velocity as speed.
• Can you think of an example of
non uniform motion?
Change in speed / velocity
• Acceleration is the quotient of change in speed and
the time interval over which the change occurs.
𝑣𝑓 − 𝑣𝑖
𝑎=
𝑡
Where
a = acceleration (m/s2)
vf = final speed (m/s)
vi = initial speed (m/s)
t = time (s)
Example 3
• A little bunny rabbit is trotting 1.5 m every second,
when it suddenly spots an angry squirrel. It then
manages to escape it by increasing its speed to 8.4
m/s in a time of 2 seconds.
• What was the little bunny rabbit’s acceleration
during that time?
Example 4
• A jet is flying at a speed of 950 km/h. When its
turbo jet fuel is then applied, it increases its speed
to 1290 km/h in 24 seconds.
• What was the jet’s acceleration during that time?
Example 5
• A badger is resting, watching peacefully the sunset,
when it suddenly hears a hamster at a not too
distant location. It immediately gets up and
accelerates toward the hopeless animal at a rate of
4 m/s2 for 6 seconds.
• What is its speed after 6 seconds?
Other NUM formulas
• All kinematics problems present a set of known
variables and unknown variables.
• You need to use the former to find the latter.
• However, the given acceleration formula only uses
initial and final speed and time. What if distance is
given with two other variables?
• Here are 3 new non uniform motion formulas…
Example 6
• A car is travelling at 12 m/s (about 45 km/h). It
approaches a stop sign and decelerates at 4.0 m/s2.
a) How long does the car take to stop?
b) How far does the car travel while stopping?
Example 7 (challenge)
In 1983, Kebede Balcha won the Montreal Marathon
(length of 42.195 km) with a time of 2 hours, 10
minutes and 9 seconds.
a) What was his average speed (in m/s)?
At the 34 km checkpoint, Balcha was ahead of the 2nd
placed runner by a time of 2.5 minutes. It was found
that, at that checkpoint, both runners were running at
the same speed. 30 seconds after Balcha crossed the
finish line, the same 2nd placed runner completed the
race.
b) What was the average acceleration of the 2nd runner
from the time Balcha passed the 34 km
checkpoint to the end?
c) What assumptions are being made here?
Problems
p.89 # 4 to 7
Free Falling Bodies
• Freefall is defined as the condition of unrestrained
motion in a gravitational field. It is characterized by
a rapid and continuing drop.
• One example of freefall can be seen with
paratroopers who jump from airplanes. Their initial
jump from the plane before opening the parachute
is known as "freefall."
Experimenting with Freefall
• Italian astronomer Galileo supposedly dropped
objects off the Leaning Tower of Pisa and showed
that they all accelerated at the same rate regardless
of mass.
• However, even today many people believe
otherwise. The fact that air resistance can
play a significant role probably contributes
to this incorrect notion.
• For objects near the surface of the Earth,
this acceleration is equal to:
g = 9.81 m/s2
So, all falling objects accelerate at
the same rate?
• Try dropping
various non living,
non expensive
objects from a
certain height. See
if certain objects
reach the ground
faster…
Introducing the Vacuum!
• Vacuum chambers are used to study certain
physical phenomena in an environment without air.
• What do you think would happen if you removed
all the air in an environment and
you dropped, lets say, a rock and
a feather at the same same
height?
• Need some convincing?
Check this out!
Terminal Velocity
• All objects dropped or
thrown from a high enough
height will experience
terminal velocity.
• It is the maximum velocity
that an object can reach.
• What factors do you think
affect terminal velocity?
Example 8
A potato is dropped from the rooftop of a building. It
takes 2.18 seconds for it to reach the ground below.
What is the height of the building?
Example 9
A rock and a sponge were dropped from a rooftop.
The rock hit the ground in 1.4 s. The sponge took 2.0 s
to fall.
a) How high was the roof?
b) Why do you think there is a difference? Explain.
Please.
Example 10
From the ground, you decide to vertically throw a
baseball as fast as you can. The speed of the ball as
you let go is measured at 29 m/s.
What was the maximum height of the baseball?