Derivation of Kinematic Equations

1
Derivation of Kinematic Equations

• View this after
• Motion on an Incline Lab

2
Constant velocity

• Average velocity equals the slope of a position
  vs time graph when an object travels at constant
  velocity.

3
Displacement when object moves with constant
velocity

• The displacement is the area under a velocity vs
  time graph

4
Uniform acceleration
This is the equation of the line of the velocity
vs time graph when an object is undergoing
uniform acceleration.
The slope is the acceleration
The intercept is the initial velocity
5
Displacement when object accelerates from rest

• Displacement is still the area under the
  velocity vs time graph. However, velocity is
  constantly changing.

6
Displacement when object accelerates from rest

• Displacement is still the area under the
  velocity vs time graph. Use the formula for the
  area of a triangle. Area ½ height times base.

7
Displacement when object accelerates from rest

• Acceleration is from the
• slope of v-t graph.
• Rearrange to get
• To find ?x, substitute for ?v in
• area formula (1/2 ht times base)

8
Displacement when object accelerates from rest

• Simplify

Assuming uniform acceleration and a starting time
0, the equation can be written
9
Displacement when object accelerates with initial
velocity

• Break the area up into two parts
• the rectangle representingdisplacement due to
  initial velocity

10
Displacement when object accelerates with initial
velocity

• Break the area up into two parts
• and the triangle representingdisplacement due
  to acceleration

11
Displacement when object accelerates with initial
velocity

• Now, Sum the two areas

Or, if starting time 0, the equation can be
written
12
Time-independent relationship between ?x, v and a

• Sometimes you are asked to find the final
  velocity or displacement when the length of time
  is not given.
• To derive this equation, we must start with the
  definition of average velocity

13
Time-independent relationship between ?x, v and a

Another way to express average velocity in
uniform acceleration is
14
Time-independent relationship between ?x, v and a

We have defined acceleration as
This can be rearranged to
and then expanded to yield
15
Time-independent relationship between ?x, v and a

Now, take the equation for displacement
and make substitutions for average velocity and
?t from the previous slides.
16
Time-independent relationship between ?x, v and a

17
Time-independent relationship between ?x, v and a

Start with
Then substitute for avg velocity and time
like shown in previous slide to get
18
Time-independent relationship between ?x, v and a

Start with
Then Simplify to get
19
Time-independent relationship between ?x, v and a

Rearrange
20
Time-independent relationship between ?x, v and a

Rearrange again to obtain the more common form
21
Which equation do I use?

• First, decide what model is appropriate
• Is the object moving at constant velocity?
• Or, is it accelerating uniformly?
• Next, decide whether its easier to use an
  algebraic or a graphical representation.

22
For Constant Velocity (or constant motion)

• If you are looking for the velocity,
• use the algebraic form
• or find the slope of the position-time graph
  (actually the same thing)

23
Constant velocity

• If you are looking for the displacement,
• use the algebraic form
• or find the area under (under is between the
  curve and the axis) the curve/line

24
Uniform acceleration

• If you want to find the final velocity,
• use the algebraic form
• If you are looking for the acceleration
• rearrange the equation above
• which is the same as finding the slope of a
  velocity-time graph

25
In Uniform acceleration

• If you want to find the displacement,
• use the algebraic form
• eliminate initial velocity and
• whole first term if the object
• starts from rest (vo Zero)
• Or, find the area under the curve

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If you dont know the time

• 1. You can solve for ?t using one of the
  earlier equations, and then solve for the desired
  quantity, or
• 2. You can use the equation
• rearranging it to suit your needs

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All the equations in one place

• constant velocity uniform acceleration

or