Derivation of Kinematic Equations

Aim: How do we solve word problems of motion? 2-5 Due Wednesday: Read textbook pages On page 111 answer questions 47, 48, 49 and 51.

GUESS !! G ivens List all givens U nknown list what you are looking for E quations write the correct equation S ubstitute with units S olve with number and correct units

Aim: How do we solve word problems of motion? 2-5 Due Wednesday: Read textbook pages On page 111 answer questions 47, 48, 49 and 51.

GUESS !! G ivens List all givens U nknown list what you are looking for E quations write the correct equation S ubstitute with units S olve with number and correct units

Word Problem Strategies 1.Know your units!Know your units! Distance, displacement, position (m) Speed, velocity (m/s) Acceleration (m/s 2 )Acceleration (m/s 2 ) Time (sec) 2.Look for key words or phrasesLook for key words or phrases Starts from rest (v i = 0 m/s)Starts from rest (v i = 0 m/s) Comes to rest, stops (v f = 0 m/s)Comes to rest, stops (v f = 0 m/s) Average speed, moving at a steady rate, constant speed (v)

3.Determine givens and unknowns from the question 4.Choose a formula from the reference table 5.Substitute into equation with numbers and units 6.Solve Algebraically, include units in your answer!

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 The displacement is the area under a velocity vs time graph

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

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 rest From slope of v-t graph Rearrange to get Now, substitute for ∆v

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 Break the area up into two parts: the rectangle representing displacement due to initial velocity

Displacement when object accelerates with initial velocity Break the area up into two parts: and the triangle representing displacement due to acceleration

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 ∆d, 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 ∆d, v and a Another way to express average velocity is:

Time-independent relationship between ∆d, v and a We have defined acceleration as: This can be rearranged to: and then expanded to yield :

Time-independent relationship between ∆d, v and a Now, take the equation for displacement and make substitutions for average velocity and ∆t

Time-independent relationship between ∆d, v and a

Simplify

Time-independent relationship between ∆d, v and a Rearrange

Time-independent relationship between ∆d, v and a Rearrange again to obtain the more common form:

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

Constant velocity If you are looking for the displacement, –use the algebraic form –or find the area under the curve

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 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… 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 constant velocityuniform acceleration

Don’t need distance Don’t need final velocity Don’t need time For constant a

What is the acceleration of a car whose speed increases from 30 m/s to 80 m/s in 10 seconds? 5 m/s 2 An object moving at 20 m/s is accelerated at 5 m/s 2. How far did it go in 15 seconds? 863m An object at rest is accelerated at 8m/s 2. What is its speed after it traveled 9m? 12 m/s

An airplane is accelerated at 5 m/s2 for 5 seconds. At this time the airplane’s velocity is 125 m/s. At what velocity did the plane start out? 100 m/s A car traveling at 30 ft/sec is brought to a stop in 6 seconds. What is the average velocity of the car? What is its acceleration? 15 ft/s -5 ft/s2