The Kinematics Equations (1D Equations of Motion) Unit 2 The Kinematics Equations (1D Equations of Motion)

Day #1 Introduction to the Equation of Motion

What are the 5 key equations we have learned thus far? A) ∆x = xf – xi finding a change in displacement B) ∆v = vf – vi finding a change in velocity C) s = d/∆t finding speed (distance/time) D) v = ∆x/∆t finding velocity (displacement/time) E) a = ∆v/∆t finding acceleration (change in velocity change in time) ¯ ¯

There are 4 new equations for motion. We’ll start with a = ∆v/∆t Substitute in ∆v = vf – vi The formula is now a = vf-vi t Rearrange to get a∙t = vf - vi ¯ Remember… slope of the line = a = vf-vi t ¯

Example #1 a = vf-vi or a∙t = vf - vi t A car starting from rest accelerates uniformly to a speed of 75 m/s in 12 seconds. What is the car’s acceleration? First, identify the known values, then plug and chug a = vf-vi or a∙t = vf - vi t = 75 m/s – 0 12s =+6.25 m/s2 Vi = 0 Vf = 75 m/s t = 12 s a = ? *always make a list of your knowns

Example #2 a = vf-vi or a∙t = vf - vi t Vf= Vi + at A spacecraft traveling at 1025 m/s is uniformly accelerated at a rate of 105 m/s2 for 9 seconds. What is the final velocity of the spacecraft? First, identify the known values, rearrange the formula, then plug and chug a = vf-vi or a∙t = vf - vi t Vf= Vi + at = 1025 m/s + 9s(105m/s2) = 1970 m/s Vi = 1025 m/s Vf = ? t = 9 s a = 105 m/s2

The second new formula Recall: area under the curve on a v/t chart gives us displacement = ∆x (or d) Consider a random, uniform accl object. Use the area of a trapezoid

Example #3 A plane is moving at a speed of 500m/hr when it lands on a runway. Accelerating uniformly, it comes to a stop after covering 15m. How long did it take to stop? Vi = 500 m/s Vf = 0 t = ? a = ? ∆x = 15 m ∆x =½∆t(vf+vi) 15 m = ½(t)(0+500 m/hr) 30 m = 500m/hr(t) t = 0.06 hr = 216 sec

Example #3 A plane is moving at a speed of 500m/hr when it lands on a runway. Accelerating uniformly, it comes to a stop after covering 15m. How long did it take to stop? What was its acceleration? Vi = 500 m/hr Vf = 0 t = 0.03 hr or 108 s (previous slide) a = ? ∆x = 15 m a = vf-vi t = 0-500m/hr 0.06 hr =-8333.33 m/hr2 Or -0.0006 m/s2

HW In book, Complete pg 53,55, & 58 odds

Day #2 RE-Introduction to the equation of motion

…. area under curve =

2. The first answer doesn’t make sense because you can’t have a negative time. You have to use the quadratic formula to solve the problem.

a = vf-vi (rearrange) t Substitute our newest formula into the derivation below

Make a decision as to which direction is POSITIVE & which is NEGATIVE List what you have (using “COMPATIBLE” units & using proper SIGNS) choose your equation Solve the equation Make sure your answer makes sense (both in MAGNITUDE & DIRECTION)

3.

4.

Review – Solve Solve the following: You roll a ball up an incline at a speed of 4.5 m/s. After 5 sec, the ball is on the way down travelling at a new speed of 1.5 m/s. Find the ball’s acceleration. (Sketch) -1.5-4.5 -6 -1.2 m/s2 5 5 rookie mistake…. 1.5-4.5 Find the time to the peak. -1.2 = 0-4.5 -1.2 = -4.5 =3.75 sec t t How far up does it go? 0 = (4.5)2 + 2(-1.2)d =8.44 m

Day #3 Free Fall

Free Fall Free fall problems are acceleration problems where we know the acceleration. For all free fall problems (at the surface of the earth), a = 9.8 m/s2 (down) the object speeds up by 9.8m/s2 after every second that it falls When solving: Make a chart Use arrows next to variables to show the direction of the vectors Fill in known variables Solve for unknowns

Free Fall Problems a = vf-vi t a = vf t Initial Velocity (Vi) will be 0, so our formulas change slightly. No need to memorize these.. a = vf-vi t a = vf t

9.8 m/s2 0 m/s 9.8 m/s 19.6 m/s 29.4 m/s 39.2 m/s 49 m/s

Free Fall Problems 1. A ball is dropped from the top of a bridge, and it takes 8 seconds to hit the water. A. How tall was the bridge? B. How fast was the ball going just before it hits the water? a. height? So find ∆x. b. a = (vf-vi)/t ∆x = vi ∆t +1/2(a)(∆t )2 9.8= vf/8 = ½(9.8)(8) 2 = 78.4 m/s = 313.6 m downward *You could solve part B first and then use the equation ∆x=1/2t(vi+vf) a 9.8m/s2 vi vf ∆x ∆t 8s

a -9.8m/s2 vi vf ∆x 20m ∆t vi= 0 a = -9.8 m/s2 Dx = -20 m Because the diver is falling DOWNWARD, we can also use a negative sign

If all objects accelerate at the same rate when falling near the earth’s surface, why do some objects actually hit the ground faster than others? (assuming they are dropped at the same time from the same height)? AIR RESISTANCE!! Some objects are narrower (like pencils) and therefore the air resists their falling less. Some things are wider (like frisbees) and the air provides more resistance and slows down their fall. NOTICE HOW THE MASS OF THE OBJECTS DOES NOT MATTER!

Notice that the acceleration is CONSTANT and NEGATIVE the ENTIRE way 5. Sometimes the variable “y” is used instead of “x” to simply show that the object is moving vertically y v t t Notice that the object speeds up Notice that the acceleration is CONSTANT and NEGATIVE the ENTIRE way

Day #4 Throw-ups, Come-downs, Throw-downs

Free Fall Problem – Throw up “A ball is thrown up from the ground…” In this type of problem, the initial velocity is always up. Also, the primary type of problem will ask “what is the maximum height”, which means vf = 0 m/s. 2. A ball is thrown up from the ground at an initial speed of 20 m/s. A. what is the maximum height the ball reaches? B. How long before it gets to the max altitude? A. ∆x =? B. ∆t =? Vf2=vi2 +2a ∆x a=(vf-vi)/ ∆ t 0=2o2 +2(-9.8) ∆x -9.8=(0-20) ∆ t ∆x = 20.4m ∆ t=2.04s *You can solve for B first then use the formula ∆x =1/2t(vi+vf) a -9.8m/s2 vi 20 m/s vf ∆x ∆t

To solve for “t”, use the quadratic formula  Problem #2 1 2 UP +, DOWN - a = -9.8 m/s2 v2 = 0 (at the top) a) vi = 20 m/s b) a = -9.8 m/s2 vf = 0 m/s c) To solve for “t”, use the quadratic formula  t= 0.66s and 3.42 sec (going up) (going down)

Throw –up/Come down Assume: Up+, Down – A =-9.8m/s2 v2= 0 (at the top) 3 scenarios 1. 2. 3. Assume: Up+, Down – A =-9.8m/s2 v2= 0 (at the top) We’ll start with scenario #3. V3=-V1

Scenario 3 – object is thrown up and caught at the same height 1 2 3 b) vf = 0 m/s a) a = -9.8 m/s2 vf = 0 m/s a = -9.8 m/s2 v i= 24.25 m/s Dx = 30 m c) Due to the fact that the ball is thrown and caught at the same height …

vf2=vi2 +2a∆x vf2=24.252 +2(-9.8)(-1) 24.65 m/s down Or-24.65m/s Scenario 2 and part d of our problem 1 2 4 3 vi= 24.25 m/s a = -9.8 m/s2 Dx = -1 m (because down is -) vf2=vi2 +2a∆x vf2=24.252 +2(-9.8)(-1) 24.65 m/s down Or-24.65m/s

Dx = -1 m (because down is -) What if we were solving for the time it was when the object hits the ground? vi= 24.25 m/s a = -9.8 m/s2 Dx = -1 m (because down is -) To solve for “t”, use the quadratic formula  Only positive times make sense

Day #5 Chase Problems

Chase Problems A chase problem is a scenario where two objects are involved, and they have the same position at some later time. They can Both start at the same place and same time Start at different places but at the same time Both start at the same place but at different times Start at different places at different times B or D but be moving in different directions

Substitute the second formula into the first, creating: The Key Equation ∆x=1/2at2+vit ∆x=xf-xi Substitute the second formula into the first, creating: x2=1/2at2+vit+xi This formula will give you the final position for both objects. Since you don’t know that final position, set the two formulas equal to each other.

1/2at2+vit+xi = 1/2at2+vit+xi Chase Problem #1 Timmy is running at a constant speed of 6m/s. He sees Susie running 50m in front of him. She is moving at a constant speed of 4 m/s, in the same direction. How long will it take for Timmy to catch Susie? Notice that while Timmy and Susie have different starting positions (xiT=0m while xiS=50m), they have the same final position (xf) after Timmy catches Susie. . . X xiT xiS x2 Tim = Susie 1/2at2+vit+xi = 1/2at2+vit+xi ½(0)t2+6t+0 = 1/2(0)t2+4t=50 6t = 4t=50 t = 25sec

1/2at2+vit+xi = 1/2at2+vit+xi Chase Problem #2 A continuation of the last problem. Timmy passes Susie, running at his constant speed of 6m/s. Susie decides to pick up the pace, very gradually. She begins to accelerate at a constant rate of 0.1m/s2. How long will it take her to catch Timmy? Notice that Timmy and Susie have the same starting position (xiT=0m xiS=0m) and ending position (xf) after Susie catches Timmy. . x xit=xis x2 Tim = Susie 1/2at2+vit+xi = 1/2at2+vit+xi ½(0)t2+6t+0 = 1/2(0.1)t2+4t+0 6t=0.05t2+4t 6=0.05t+4 t=40sec Really, you could have made the initial position anything you wanted – they cancel out. I decided to choose 0.

Chase Problem #3 A man drops a penny off the top of a 100m tall building. Exactly 1 second later another man throws a nickel downward from the same place as the first man. What is the minimum speed the nickel must be thrown at in order to catch the penny? Start with the time it takes the penny to hit the bottom using the formula: ∆x=1/2at2+vit -100=1/2(-9.8)t2+0t t= 4.52 sec penny = nickel (1 sec later) 1/2at2+vit+xi = 1/2at2+vit+xi ½(-9.8)(4.52)2+0(4.52)+0 = 1/2(-9.8)(3.52)2+vi(3.52)+0 Vi= -11.19 m/s (neg. because it’s going down) + -

Chase Problem #4 really hard In a strange, yet exciting, crash-test-dummy crash, two cars start by facing each other 1000m apart on a straight road. The first car accelerates from rest with a constant accel of 4m/s2. The second car accel, at a rate of 8 m/s2 for 5 sec but then settles into a constant speed. Find the elapsed time before these two cars collide. Notice that the dummies have different starting positions (xiA=0, xiB=1000m), but they have the same final position (x2) when they crash. Start by finding the final velocity of car B by using the following formula: a = vf-vi -8=(vf-0)/5 so vf =-40m/s t 1/2at2+vit+xi = 1/2at2+vit+xi

Test Review

Write Down What You Have (Look for “Key” Words) Step #1 Write Down What You Have (Look for “Key” Words)

“Coming to a stop” “Starting from rest” “Coasting” “Maximum Height” “Dropped” vf = 0 vi = 0 vi = vf = constant vf = 0 a = -9.8 m/s2

“Slowing Down” “Braking” “Speeding up” “Accelerating from rest”

Make sure your answer makes sense Step #3 Solve the Equation Step #4 Make sure your answer makes sense

Choose your “Key Points” in every problem…and do so wisely. Helpful Tip #1 Choose your “Key Points” in every problem…and do so wisely.

“Free-fall” “Throw up” Vertical Problems “Throw-downs” “Free-fall” “Throw up” 1 vi = 0 1 vi 0 2 2 2 vf = 0 (at top) 1

“Throw up / Come Down” (throw and catch at same height) Vertical Problems “Throw up / Come Down” (throw and catch at same height) 2 vf = 0 (at top) v1 = -v3 1 3

“Throw up / Come Down” (throw and catch at different heights) Use and solve quadratically for “t” 2 1 3 2 1 3 Dx13 = + Dx13 = -

Assign positive and negative to different directions. Helpful Tip #2 Assign positive and negative to different directions.

When solving a quadratic equation, do so with minimal effort. Helpful Tip #3 When solving a quadratic equation, do so with minimal effort.

Solving a quadratic equation Choice A Choice B Factoring Unlikely on a physics problem  Choice C  2nd Trace  Zero (on graphing calculator)

Can be used at constant speeds (a=0) or when accelerating. Awesome Dude!

Chase Problems Since the two objects (A and B) end up at the same position by the end of the chase, use …

It’s already accounted for here and here  But what if… The objects start at different places? It’s already accounted for here and here 

But what if… The objects start at different TIMES? You’ll need to use an extra equation relating the two times. Plug this new equation into the long equation above. Example: tA = tB + 1

Helpful Tip #6 It is always important to remember that when something is thrown up or down (or simply falls), the acceleration at ALL times is constant.

The acceleration of the ball at EVERY point on this red path (When it’s rising up, when it’s stopped, when it’s falling down) is always -9.8 m/s2. + -

An object thrown up has a constant acceleration at ALL times…. ….the acceleration due to gravity x Objects rise and fall in the same amount of time (assuming no parachute ) t v t Constant slope = constant accel.

Complete Review Worksheet TONIGHTS HW Complete Review Worksheet