Download Kinematics Equation Lecture


Kinematic Equations


the branch of mechanics concerned with the motion
of objects without reference to the forces that cause
the motion.
Two types of Equations


Constant Velocity Equations (acceleration = 0)
Constant Acceleration questions (velocity is
constantly changing) (a≠0 but is a constant…the
same)

Δv= Δx/Δt (y instead of x for vertical)

Δvavg=xtotal/Δt total (book uses 𝒗 for avg velocity)
 Note: constant velocity means

xf=xi+vavgΔt (y for vertical)
vinst =vavg
•
Recall that for an object moving at a constant velocity,
displacement is equal to average velocity times the time
interval.
x=vt

For an object that is changing velocity and uniformly
accelerating, the average velocity can be written.
𝒗𝒇 + 𝒗𝒊
𝒗=
𝟐
Therefore: x=
𝑽𝒇+𝑽𝒊
t
𝟐


If an object’s average acceleration during a
time interval is known, the change in
velocity during that time can be found.
The definition of average acceleration
𝜟𝒗
𝒗𝒇−𝒗𝒊
ā= =
𝜟𝒕

𝜟𝒕
Solving for final velocity:
𝒗𝒇 = 𝐯𝐢 + ā 𝜟t

In cases in which the acceleration is constant,
the average acceleration, , is the same as the
instantaneous acceleration, a. The equation for
final velocity can be rearranged to find the time
at which an object with constant acceleration
has a given velocity.
𝒗𝒇 = 𝐯𝐢 + 𝐚 Δt

If the vi , a and Δt are known, the displacement
can be found by combining….
Vf=vi+aΔt

𝒙=
𝑽𝒇+𝑽𝒊
𝟐
Δt
Substituting the equation for vf into the 2nd
equation results in:(I did not use the Δ for t but
its there!!! Book does this all the time)
x= 𝒗𝒊𝒕 +
𝟏
𝒂𝒕𝟐 or
𝟐
xf = x0 + 𝒗𝒊𝒕 +
𝟏
𝒂𝒕𝟐
𝟐

Solve this equation vf = vi+ a∆t for time….
∆t=

𝒗𝒇+𝒗𝒊
𝒂
Substituting the equation for “t” into
x=
𝑽𝒇+𝑽𝒊
∆t
𝟐
RESULTS IN:
𝟐
𝒗𝒇
=
𝟐
𝒗𝒊
+ 𝟐𝒂∆x
(this is known as the “timeless equation”)
𝐕𝐟+𝐕𝐢
∆t
𝟐

𝐱=

𝐯𝐟 = 𝐯𝒊 + 𝐚∆t

xf= 𝐱𝐨 + 𝐯𝒊∆𝐭 +

𝒗𝟐𝒇 = 𝒗𝟐𝒊 + 𝟐𝒂∆𝒙

𝟏
𝐚∆𝐭𝟐
𝟐
I use x and y depending on horizontal or vertical. Many
times the ∆ is missing but its there!!!
© 2014 Pearson Education,
Inc.
1.
As you drive in your car at 15 m/s (just a bit under 35
mph), you see a child’s ball roll into the street ahead of
you. You hit the brakes and stop as quickly as you can.
In this case, you come to rest in 1.5 s. How far does
your car travel as you brake to a stop?
• Draw the pictures (particle motion and
V-T)
• Do not forget the v and a vectors
Question 1 Answer
ACCELERATION:
Distance Traveled:
© 2015 Pearson Education, Inc.
Question 2: Kinematics of a rocket launch
A Saturn V rocket is launched straight up with a constant acceleration of 18 m/s2.
After 150 s, how fast is the rocket moving and how far has it traveled?
© 2015 Pearson Education, Inc.
Question 2: Kinematics of a rocket launch
(cont.)
Speed:
Distance Traveled:
© 2015 Pearson Education, Inc.
Question 3: Calculating the minimum length of
a runway
A fully loaded Boeing 747 with all engines at full thrust accelerates at 2.6
m/s2. Its minimum takeoff speed is 70 m/s. How much time will the plane
take to reach its takeoff speed? What minimum length of runway does the
plane require for takeoff? USE YOUR WORKSHEET
DRAW THE PICTURE!!!!!
Question 3: Calculating the minimum length of
a runway (cont.)
Time:
© 2015 Pearson Education, Inc.
Question 3: Calculating the minimum length of
a runway (cont.)
Runway Length:
© 2015 Pearson Education, Inc.

A position-time graph of a bike
moving with constant
acceleration is shown on the
right. Which statement is correct
regarding the displacement of
the bike?
A. The displacement in equal time interval is constant.
B. The displacement in equal time interval
progressively increases.
C. The displacement in equal time interval
progressively decreases.
D. The displacement in equal time interval first
increases, then after reaching a particular point it
decreases.
Example Problem (if We have time): Champion
Jumper
The African antelope known as a
springbok will occasionally jump straight
up into the air, a movement known as a
pronk. The speed when leaving the ground
can be as high as 7.0 m/s.
If a springbok leaves the ground at 7.0 m/s:
A. How much time will it take to reach its highest point?
B. How long will it stay in the air?
C. When it returns to earth, how fast will it be moving?
Answers:
A: 0.71 s
B: 1.4 s
C: 7.0 m/s