Derivation of kinematic equations
Download
1 / 27

Derivation of Kinematic Equations - PowerPoint PPT Presentation


  • 491 Views
  • Updated On :

Derivation of Kinematic Equations. View this after Motion on an Incline Lab. Constant velocity. Average velocity equals the slope of a position vs time graph when an object travels at constant velocity. Displacement when object moves with constant velocity.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Derivation of Kinematic Equations' - Ava


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Derivation of kinematic equations l.jpg

Derivation of Kinematic Equations

View this after

Motion on an Incline Lab


Constant velocity l.jpg
Constant velocity

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


Displacement when object moves with constant velocity l.jpg
Displacement when object moves with constant velocity

The displacement is the area under a velocity vs time graph


Uniform acceleration l.jpg
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


Displacement when object accelerates from rest l.jpg
Displacement when object accelerates from rest

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


Displacement when object accelerates from rest6 l.jpg
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.


Displacement when object accelerates from rest7 l.jpg
Displacement when object accelerates from rest

From slope of v-t graph

Rearrange to get

Now, substitute for ∆v


Displacement when object accelerates from rest8 l.jpg
Displacement when object accelerates from rest

Simplify

Assuming uniform acceleration and a starting time = 0, the equation can be written:


Displacement when object accelerates with initial velocity l.jpg
Displacement when object accelerates with initial velocity

Break the area up into two parts:

the rectangle representingdisplacement due to initial velocity


Displacement when object accelerates with initial velocity10 l.jpg
Displacement when object accelerates with initial velocity

Break the area up into two parts:

and the triangle representingdisplacement due to acceleration


Displacement when object accelerates with initial velocity11 l.jpg
Displacement when object accelerates with initial velocity

Sum the two areas:

Or, if starting time = 0, the equation can be written:


Time independent relationship between x v and a l.jpg
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:


Time independent relationship between x v and a13 l.jpg
Time-independent relationship between ∆x, v and a

Another way to express average velocity is:


Time independent relationship between x v and a14 l.jpg
Time-independent relationship between ∆x, v and a

We have defined acceleration as:

This can be rearranged to:

and then expanded to yield :


Time independent relationship between x v and a15 l.jpg
Time-independent relationship between ∆x, v and a

Now, take the equation for displacement

and make substitutions for average velocity and ∆t






Time independent relationship between x v and a20 l.jpg
Time-independent relationship between ∆x, v and a

Rearrange again to obtain the more common form:


Which equation do i use l.jpg
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 it’s easier to use an algebraic or a graphical representation.


Constant velocity22 l.jpg
Constant velocity

If you are looking for the velocity,

  • use the algebraic form

  • or find the slope of the graph (actually the same thing)


Constant velocity23 l.jpg
Constant velocity

  • If you are looking for the displacement,

    • use the algebraic form

    • or find the area under the curve


Uniform acceleration24 l.jpg
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


Uniform acceleration25 l.jpg
Uniform acceleration

If you want to find the displacement,

  • use the algebraic form

  • eliminate initial velocity if the object starts from rest

  • Or, find the area under the curve


If you don t know the time l.jpg
If you don’t know the time…

You can solve for ∆t using one of the earlier equations, and then solve for the desired quantity, or

You can use the equation

  • rearranging it to suit your needs


All the equations in one place l.jpg
All the equations in one place

constant velocity uniform acceleration