Visualizing kinematics: the real reason graphs are important

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. AKA: intro to derivatives and integrals

Remember: Constant Acceleration formulas 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 v = v0 + at Δx = ½(v+v0)t x = x0 + v0t + ½ at2 v2 = v02 + 2a(Δx)

Part 1: the derivative • You know: the slope of the line on a graph tells you something…. Object 1: positive slope = positive velocity Object 2: zero slope= zero velocity Object 3: negative slope = negative velocity Click on Graph for Flash Animation

Instantaneous acceleration The instantaneous acceleration is the acceleration of an object at some instant or at a specific point in the object’s path. 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. velocity a=12 m/s2

Tells you the….. The slope of the tangent line at a point on a….. • d vs. t graph…. • v vs. t graph…. • Velocity at any “t” • 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.

The derivative is the slope of the original function.

“the derivative of f with respect to x” “f prime x” or “y prime” “the derivative of y with respect to x” or “dee why dee ecks” “the derivative of f with respect to x” or “dee eff dee ecks” “the derivative of f of x” “dee dee ecks uv eff uv ecks” or

Differentiation Rules!! Note: These are only the most basic. Mr. Norman will have a more complete list and much better explanations than I can 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: f(x) = 2

Rule 2: Power rule Example: f(x) = x3 Example: F’(x) = 3x2 Example: f(x) = x Example: F’(x) = 1

Rule 3: constant multiple rule Example: f(x) = 2x3 Example: F’(x) = 6x2 Example: f(x) = 3x Example: F’(x) = 3

Rule 4: sum and difference rule Example: f(x) = 2x3 + 5x2 Example: F’(x) = 6x2 + 10x Example: f(x) = 3x3 + x2 +3 Example: F’(x) = 9x2 + 2x

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? f(x) = 5x3 + x2 -1 • f’(x)=15x2 + 2x • f’(x)= 15x2 + 2x -1 • f’(x)= 2x2 + x - 1 • f’(x)= Impossible to determine

Ironman’s position as a function of time is defined as x=t2 + 4. What is his velocity after 6 seconds? • 16 m/s • 52 m/s • 40 m/s • 3 m/s • 12 m/s

Ironman’s position as a function of time is defined as x=t2 + 4. What is his acceleration after 6 seconds? • 2 m/s2 • 3 m/s2 • 6 m/s2 • 0.5 m/s2 • 1 m/s2

The silver surfer’s acceleration is defined by the following function: a=3t +5t2. What is his velocity after 10 seconds? • 1817 m/s • 532 m/s • 510 m/s • 1950 m/s • Impossible to determine

Integral Calculus!! Physic’s most powerful 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? • five • twelve • fourteen • Thirty-one • 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 graph represents the distance you have traveled!!

This is useful: you can do the same thing with graphs of a vs t Remember this for calculus: the simplest definition of an integral = the area under a curve! Example: A particle accelerates at 5 m/s2 for 30 s. acceleration m/s2 What would the area of this rectangle represent? 5 m/s2 30 (s) m m Time (s)  s = s2 s The area under an “a vs t” graph is the velocity!

The Flash goes for a walk. His speed vs time graph is shown below. Two hours into the trip how fast is he going? • Zero mph • 10 mph • 20 mph • 30 mph • 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? • 40 miles • 80 miles • 110 miles • 120 miles • 210 miles

The Flash starts from rest and accelerates to 60 mph in 10 s. How far does he travel during those 10 seconds? • 1/60 mile • 1/12 mile • 1/10 mile • ½ mile • 60 miles

velocity 60 mph 10 seconds time

Visualizing kinematics: the real reason graphs are important