Visualizing kinematics: the real reason graphs are important

• What if we have Non-constant acceleration? Can we still predict velocities and displacements?

• In order to really understand this at a higher level you need to be able to visualize the math in a new way.

• Remember that calculus is the blending of Algebra and geometry.

Remember: Constant

Acceleration formulas

v = v0 + at

Δx = ½(v+v0)t

x = x0 + v0t + ½ at2

v2 = v02 + 2a(Δx)

These are used to solve problems with CONSTANT acceleration (ex: free fall). You are usually given 3-5

quantities and you need to find the

rest

Object 1: positive slope = positive velocityObject 2: zero slope= zero velocity Object 3: negative slope = negative velocity

Click on Graph for Flash Animation

Part 1: the derivative• You know: the slope of the line on a graph tells you something….

Instantaneous acceleration

The instantaneous acceleration at a given time can be determined by measuring the slope of the line that is tangent to that point on the velocity-versus-time graph.

The instantaneous acceleration is the acceleration of an object at some instant or at a specific point in the object’s path.

velo

cit

y a=12 m/s2

The slope of the tangent line at a point

on a…..

1. d vs. t graph….

2. v vs. t graph….

Tells you the…..

1. Velocity at any “t”

2. Acceleration at any “t”

Wouldn’t it be great if there was an “easy” way to find the slope at any point on our graph?!?

• There is!!

• And there are a bunch of hard ways to do it!! –I’ll leave those for Mr. Norman to explain.

• The derivative of a function provides a function for the slope of the original function.

y f x

y f x

The derivative is the slope of the original

function.

f x “f prime x” or “the derivative of f with respect to x”

y “y prime”

dy

dx“dee why dee ecks” or “the derivative of y with

respect to x”

df

dx“dee eff dee ecks” or “the derivative of f with

respect to x”

df x

dx“dee dee ecks uv eff uv ecks” or “the derivative

of f of x”( of of )d dx f x

Note: These are only Note: These are only the most basic. Mr. the most basic. Mr. Norman will have a Norman will have a

more complete list and more complete list and much better much better

explanations than I can explanations than I can give.give.

Rule 1: a constant

If the derivative of a function is its slope, then for a constant function, the derivative must be zero.

Example: Example:

f(x) = 2f(x) = 2

Rule 2: Power rule

Example: Example:

f(x) = xf(x) = x33

1n ndx nx

dx

Example: Example:

F’(x) = 3xF’(x) = 3x22

Example: Example:

f(x) = xf(x) = x

Example: Example:

F’(x) = 1F’(x) = 1

Rule 3: constant multiple rule

Example: Example:

f(x) = 2xf(x) = 2x33

Example: Example:

F’(x) = 6xF’(x) = 6x22

Example: Example:

f(x) = 3xf(x) = 3x

Example: Example:

F’(x) = 3F’(x) = 3

d ducu c

dx dx

Rule 4: sum and difference rule

Example: Example:

f(x) = 2xf(x) = 2x3 3 + + 5x5x22

Example: Example:

F’(x) = 6xF’(x) = 6x2 2 + + 10x10x

Example: Example:

f(x) = f(x) = 3x3x33 + x + x22 +3+3

Example: Example:

F’(x) = 9xF’(x) = 9x22 + + 2x2x

d du dvu v

dx dx dx d du dv

u vdx dx dx

Rule 5: Product rule

Rule 6: Quotient rule

I’ll leave for later…

What you need to be able to do…

• Find the derivatives of functions using the previous rules.

• Find the original function if given a derivative.

• Understand what derivatives are and what they tell us.

• Remember:– dx/dt = v– dv/dt = a

What is the derivative of the following function?

A. f’(x)=15x2 + 2x

B. f’(x)= 15x2 + 2x -1

C. f’(x)= 2x2 + x - 1

D. f’(x)= Impossible to determine

f(x) = 5x3 + x2 -1

Ironman’s position as a function of time is defined as x=t2 + 4. What is his velocity after 6 seconds?

A. 16 m/sB. 52 m/sC. 40 m/sD. 3 m/sE. 12 m/s

Ironman’s position as a function of time is defined as x=t2 + 4. What is his acceleration after 6

seconds?A. 2 m/s2

B. 3 m/s2

C. 6 m/s2

D. 0.5 m/s2

E. 1 m/s2

The silver surfer’s acceleration is defined by the following function: a=3t +5t2. What is his

velocity after 10 seconds?

A. 1817 m/s

B. 532 m/s

C. 510 m/s

D. 1950 m/s

E. Impossible to determine

Physic’s Physic’s most most powerful powerful tool.tool.

Suppose you are going on a long bike ride. You ride one hour at five miles per hour. Then three hours at four miles per hour and then two hours

at seven miles per hour. How many miles did you ride?

A. five

B. twelve

C. fourteen

D. Thirty-one

E. Thirty-six

You just used arithmetic to find the answer. Arithmetic is blind. It has uses but if you get too complex

you will lose sight of what you are doing.

Lets look at it with geometry. It has “eyes” and is easy to visualize. Graph the problem (velocity vs. time).

The area under this The area under this graph represents the graph represents the distance you have distance you have traveled!!traveled!!

This is useful: you can do the same thing with graphs of a vs t

Time (s)

accele

rati

on

m/s

2

Example: A particle accelerates at 5 m/s2 for 30 s.

30 (s)

5 m

/s2 What would the

area of this rectangle represent?

m

s2 s =

m

s

The area under an “a vs t” graph is the velocity!

Remember this for calculus: th

e simplest

Remember this for calculus: th

e simplest

definition of an integral = the area under a

definition of an integral = the area under a

curve!curve!

The Flash goes for a walk. His speed vs time graph is shown below.

Two hours into the trip how fast is he going?

A. Zero mph

B. 10 mph

C. 20 mph

D. 30 mph

E. 40 mph

The Flash goes for a walk. His speed vs time graph is shown below.

How far did he go during the whole trip?

A. 40 miles

B. 80 miles

C. 110 miles

D. 120 miles

E. 210 miles

The Flash starts from rest and accelerates to 60 mph in 10 s. How far does he travel during

those 10 seconds?

A. 1/60 mile

B. 1/12 mile

C. 1/10 mile

D. ½ mile

E. 60 miles

time

60 m

ph

velo

cit

y

10 seconds