Picturing Motion on a Cartesian Graph You can

Picturing Motion on a Cartesian Graph

You can also describe motion with a Velocity vs Time Graph

Car “A” Car “B” Describe what is occurring Car “A” is moving at a constant velocity of +11 m/s Car “B” is moving at a constant velocity of -33 m/s

positive velocity (slowing) Positive velocity (constant) Describe the motion

Positive velocity( slowing) stopped Negative velocity, (speeding up) Describe the motion Is the acceleration positive or negative? negative

Stopped (0 m/s) negative velocity (slowing) Describe the motion

positive velocity (speeding up) Describe what is occurring What is the slope of this line? 10 m/s - 0 m/s =. 25 m/s 2 40 s - 0 s

The slope of a V vs T 10 m/s - 0 m/s 2 =. 25 m/s graph is the acceleration. 40 s - 0 s Note that the acceleration is POSITIVE

Is the acceleration constant? Yes, because the slope is constant!!!

What does this graph tell you about the two objects?

Applets for graphs 2. 5

Can do WS 2. 3 (Motion Graph Interpretation WS) The one with lots of pictures of graphs

Positive velocity, speeding up at an exponentially Describe the motion

What is the average acceleration over 90 seconds? a= v 2 - v 1 t 2 - t 1 = 48 m/s - 3 m/s 90 s - 0 s =. 50 m/s 2

How would you find instantaneous acceleration? Tangent Lines

Finding Displacement from a velocity time graph. Dx v= Dt Dx = Dt v

50 m/s The area under a V vs T line is the displacement 30 s Find the displacement from time = 40 seconds to time = 70 sec Dx = Dt v = 50 m/s 30 s

v 2 = 30 m/s v 1 = 0 m/s Find the displacement from time = 0 s to 30 sec Using the equation below Dx = Dt v And by finding the area under the line

+900 m -1200 m What is the displacement from 0 – 70 s? -300 m

Could we find the displacement here? Just break it up into smaller problems

OR

v 2 v 1 t 2 Describe the scenario Find the displacement between the two points

v 2 Area 2 v 1 Area 1 t 2 Find the displacement between the two points Displacement = Area 1 + Area 2

v 2 Area 2 v 1 Area 1 t 2 Area 1 = V 1(t 2 -t 1) = v 1 Dt

v 2 height Area 2 v 1 base v 1 Dt t 1 Area 2 = 1/2 base * height = 1/2 (t 2 -t 1) (v 2 -v 1) = 1/2 Dt (v 2 -v 1) t 2

v 2 1/2 Dt (v 2 -v 1) v 1 height base v 1 Dt t 1 t 2 Displacement = v 1 Dt + 1/2 Dt (v 2 -v 1)

Displacement = v 1 Dt + 1/2 Dt (v 2 -v 1) Dx = Dt(v 1 + 1/2 (v 2 -v 1)) Dx = (v + 1/2 (v -v )) 1 2 1 Dt Dx = v + 1/2 v - 1/2 v 1 2 1 Dt Dx = v - 1/2 v + 1/2 v 1 1 2 Dt

Dx = v - 1/2 v + 1/2 v 1 1 2 Dt Dx = 1/2 v + 1/2 v 1 2 Dt Dx = 1/2 (v + v ) 1 2 Dt Dx (v + v ) 2 v= 1 = Dt 2

This works for any situation where a triangle is formed our shapes fit the line v v t t What is different about acceleration?

(v + v ) 1 2 v= Dx Dt Shortcut ONLY WORKS if acceleration is constant. Meaning velocity is a straight line. Is always TRUE!!! It is true by definition.

What is the average velocity from 0 to 60 seconds?

How do would you find Initial position displacement from 20 to 60 seconds instantaneous speed at 30 seconds average speed from 0 to 40 seconds acceleration

How do would you find Initial position displacement from 20 to 60 seconds instantaneous speed at 50 seconds average speed from 0 to 70 seconds average acceleration from 10 to 90 seconds

How do would you find Initial position displacement from 20 to 60 seconds instantaneous speed at 50 seconds average speed from 0 to 70 seconds average acceleration from 10 to 90 seconds

Initial position displacement from 0 to 80 seconds average speed from 0 to 80 seconds instantaneous speed at 80 seconds Acceleration from 0 to 80

Initial position displacement from 0 to 90 seconds average speed from 0 to 90 seconds instantaneous speed at 90 seconds Acceleration from 0 to 90

Motion Graph WS Calculations with motion graphs WS

Equations SO FAR Dx v = Dt (v 2 + v 1) v= 2 a= Dv Dt

A car accelerates uniformly from rest to 48 m/s in 8. 9 seconds. How far did it travel in this time? Dx v = Dt (v 2 + v 1) v= 2 Dv a= Dt

A roller coaster starts from rest and accelerates at 6. 8 m/s 2 for 18 seconds. How far does it travel? Dx v = Dt (v 2 + v 1) v= 2 Dv a= Dt

A car initially moving at 13 m/s accelerates and travels 120 m in 4. 5 seconds. What was its final speed? (assuming a constant acceleration) Dx v = Dt (v 2 + v 1) v= 2 Dv a= Dt

A car starts from rest and reaches 60 m/s in 45 meters. What was its acceleration? Dx v = Dt (v 2 + v 1) v= 2 Dv a= Dt

What was the initial speed of a car which finishes a 600 m race at 65 m/s, if it did so in 38 seconds? Dx v = Dt (v 2 + v 1) v= 2 Dv a= Dt

A car starts from rest and accelerates at 3. 5 m/s 2. How far has it gone when it reaches 25 m/s? Dx v = Dt (v 2 + v 1) v= 2 Dv a= Dt

Motion Problems WS

Derivation Dx v = Dt (v 2 + v 1) v= 2 a= Dv Dt Dx = v 1 t + at 2 2

Dx = v 1 t + Displacement(m) 2 at 2 Time (s) Change in time Initial velocity(m/s) Acceleration (m/s 2)

What acceleration would cause a car starting from a dead stop at 220 m to reach 110 m in 30. 0 seconds? Dx = v 1 t + 2 at 2 30 s 110 m 220 m a = ? ? v 0 = 0 m/s d 0 = 220 m df = 110 m t = 30 s

at 2 Dx = v 1 t + 2 Applets for graphs Exploration 2. 4

The last big one, which variable is left out? v 2 = v 1 + 2 a Dx 2 2 Final Velocity(m/s) Acceleration (m/s 2) Initial velocity(m/s) displacement (m)

A car accelerates at a constant rate of 2. 5 m/s 2 and finishes an 810 m straight track and reaches a final velocity of 64 m/s. What was its initial velocity? v 2 = v 1 + 2 a(x 2 - x 1) 2 2 vo = ? vf = 64 m/s 0 m 810 m a = 2. 5 m/s 2 df = 800 m do = 0 m vf = 64 m/s vo = ? ? ?

Dx Dv v = Dt a = Dt (v 2 + v 1) ofvequations = 2 at 2 Dx = v 1 t + 2 v 2 = v 1 + 2 a Dx 2 2 YOUR ARSENAL of equations a is average acceleration ( constant) t is change in time UNITS NEED to match

A person drops a ruler from rest which Dx Dv 2. The ruler is ofv equations accelerates at 9. 8 m/s = a= Dt Dt caught after 15. 3 cm of travel. How long (v 1 + v 2) ofv equations did it take the person to catch it? = 2 Dx = v 1 t + at 2 2 v 22 = v 21 + 2 a. Dx a = 9. 8 m/s 2 x 2 = 15. 3 cm =. 153 m x 1 = 0 m t = ? ? v 1 = 0 m/s

Dx ofv equations = a= Dt (v 1 + v 2) ofv equations = 2 Dx = v 1 t + at 2 2 v 22 = v 21 + 2 a. Dx Dv Dt A car starts at 74 m with an initial velocity of -35 m/s and accelerates at 6. 5 m/s 2 for 23 seconds. What is its final position?

Dx ofv equations = a= Dt (v 1 + v 2) ofv equations = 2 Dx = v 1 t + at 2 2 v 22 = v 21 + 2 a. Dx Dv Dt What is the final speed of a meteorite which enters our atmosphere at 150 m/s and accelerates at -4. 6 m/s 2 through the atmosphere which is 1500 m thick.

How long will it take to travel 524 m at an Dx Dv ofv equations average velocity of 12. 4 m/s = a= Dt Dt (v 1 + v 2) ofv equations = 2 Dx = v 1 t + at 2 2 v 22 = v 21 + 2 a. Dx

Dx ofv equations = a= Dt (v 1 + v 2) ofv equations = 2 Dx = v 1 t + at 2 2 v 22 = v 21 + 2 a. Dx Dv Dt A car starts from rest and accelerates to 15. 6 m/s in 61 seconds how far did it travel in this time? v 0 = 0 m/s vf = 15. 6 m/s t = 61 s df = ? ? ? d 0 = 0 m Or use could have used

Problems page 43 19 – 26, 29 19 20 21 22 23 24 2. 2 m/s 2, 110 m -2. 4 m/s 2 150 m 4. 41 m/s 2, 2. 61 s 62. 5 m 33 m/s I would try reading and trying the question on page 41 1 -13 (these are like multiple choice questions) 25 160 m, 25 s, 12 m, 10 m 26 -390 m/s 2 29 24. 0 s , 240 km/h Read section 2 -7

Gravity accelerates ALL objects in free fall at g = 9. 80 2 m/s If gravity is the only force acting on an object (no wind resistance etc…)

1 kg and 5 kg hunks of lead are dropped from a building Which hits the ground first?

If they both accelerate at 9. 8 m/s 2 they hit at the same time!!!

But does this really happen? ? ? Do heavy object fall at the same rate as light ones? ? ?

Objects accelerate the same NO matter the mass? If I drop a rock and piece of paper will they fall at the same rate?

Why don’t all things fall at the same rate on earth Air resistance

If we could drop stuff in an environment without air we could test this theory. Lunar drop

Gravity accelerates objects in free fall at g = 9. 80 2 m/s Should g be positive or negative?

Freefall speed of objects dropped from rest Time Falling (s) Instantaneous Speed (m/s) 0 1 2 3 0 9. 8 19. 6 29. 4 9. 8 t t

Whatiswould the velocity of anwhen object in What the initial acceleration freefallfrom lookrest? like graphed dropped

0 s How fast is the rock moving after 1 second? 0 m/s 1 s 9. 8 m/s 2 s 19. 6 m/s Are these its average velocity or Instantaneous velocity 3 s 4 s 29. 4 m/s 30. 2 m/s

0 s How far did the rock fall during the 1 st second of freefall 0 m/s 1 s 9. 8 m/s 2 s 19. 6 m/s 3 s 29. 4 m/s 30. 2 m/s 4 s 40 m/s

0 s 0 m/s 1 s Why does the distance traveled each second increase each second? 9. 8 m/s 5 m 15= m ? 2 s 19. 6 m/s = ? m 25 3 s 29. 4 m/s 35 = ? m 4 s 30. 2 m/s

Distance fallen changes with t 2

How much does the velocity of an object increase each second of fall? How much does the Displacement of an object increase each second of fall?

v = gt gt 2 Dy = 2

When working “gravity” problems acceleration is a given (9. 80 m/s 2) There are 3 main types of problems Starting from rest Throwing down Throwing upward

If I drop a rock (from rest) off a building, what is its velocity after 2. 5 seconds? How far has it moved? Draw a picture v 1 = 0 m/s Define coordinates v 2 = ? ? ? Establish what you know Pick the equation(s) t = 2. 5 sec a = 9. 80 m/s 2 + Dv = at Dv = 9. 80 m/s 2 2. 5 s v 2 = 24 m/s How far has it traveled?

If I drop a rock (from rest) off a building, what is its velocity after 2. 5 seconds? How far has it moved then? v 1 = 0 m/s v 2 = 25 m/s t = 2. 5 sec a = 9. 80 m/s 2 How far has it traveled? 1 way to solve is: Find average velocity + v= 0 m/s + 24 m/s = 12 m/s 2 You now know average velocity and time Dy = v Dt = 12 m/s 2. 5 s = 30 m

How far has it traveled? Another strategy Dy = v 1 t + 2 at 2 v 1 = 0 m/s t = 2. 5 sec a = 9. 80 m/s 2

A rock is dropped from rest, 6. 0 meters above the ground. How fast is it going when it hits the ground? Draw a picture Define coordinates Establish what you know Pick the equation(s) Dy ofv equations = a= Dt 6 m + Dv Dt (v 1 + v 2) ofv equations = 2 Dy= v 1 t + at 2 2 v 22 = v 21 + 2 a(y 2 - y 1) v 1 = 0 m/s y 2 = 6 m y 1 = 0 m a = 9. 80 m/s 2 vf = ? ?

A rock is dropped from rest, 6. 0 meters above the ground. How fast is it going when it hits the ground? v 0 = 0 m/s y 2 = 6 m y 1 = 0 m a = 9. 80 m/s 2 vf = ? ? Draw a picture Define coordinates Establish what you know Pick the equation(s) 6 m 2 + 2 vf = vo + 2 a(y 2 – y 1) vf 2 = (0 m/s)2 + 2 · 9. 80 m/s 2 · 6. 0 m vf = 11 m/s (down)

You drop a rock (from rest) and hear it hit 4. 6 s later. How high are you up? v 1 = 0 m/s + Dy ofv equations = a= Dt Dv Dt (v 1 + v 2) ofv equations = 2 y 2 = y 1 + v 1 t + at 2 2 v 22 = v 21 + 2 a(y 2 - y 1) t = 4. 6 sec a = 9. 80 m/s 2 y 2 = ? ? ? y 1 = 0 m

You drop a rock (from rest) and hear it hit 4. 6 s later. How high are you up? v 0 = 0 m/s t = 4. 6 sec a = 9. 80 m/s 2 df = ? ? ? d 0 = 0 m + y 2 = y 1 + vot + at 2 2 y 2 = 0 m + 0 m/s 4. 6 s + 9. 8 m/s 2 (4. 6 s)2 y 2 = 210 m

If you want to know high you are. Just drop an rock. Count 1 Mississipi, etc… your height (in m) is just y= Dy= v 1 t + 9. 8(time)2 2 if you don’t have a calculator approximate You drop and rock and count 5 Mississippi’s How high are you? about 250 meters ( about 750 feet) 2 at 2

Throwing downward Will the object speed up or slow down? v 1 g What are the signs on v 1 and g?

If you throw a rock down at 6. 4 m/s from a height of 34 meters. How fast is it going when it hits the ground? Dx ofv equations = a= Dt Dv Dt (v 1 + v 2) ofv equations = 2 x 2 = x 1 + v 1 t + at 2 2 v 22 = v 21 + 2 a(y 2 - y 1)

Throwing upward Will the object speed up or slow down? vo g What are the signs on v 1 and g?

If I toss a rock upwards at 30 m/s, what will happen to its speed? For simplicity, I am going to round gravity to 10 m/s 2 0 m/s 3 s 2 s 4 s 10 m/s 1 s 20 m/s 30 m/s 40 m/s 5 s 6 s 7 s

0 m/s The speed is the same when it hits the ground as…. . Is the velocity the same as when it left? 3 s 2 s 4 s 10 m/s 1 s 20 m/s 30 m/s 5 s 6 s What does this tell you about shooting a gun into the air?

What is the acceleration here? 0 m/s 3 s 2 s 4 s 10 m/s 1 s 20 m/s 30 m/s 5 s 6 s

For an object thrown upward Where is the highest point? What is its acceleration then?

Key things to remember about this. The speed at the highest point is…. 0 m/s 3 s 2 s The speed when it reaches the ground again is… 4 s 10 m/s 1 s 20 m/s 5 s The time it takes to fall back down is equal to…. 30 m/s 6 s The time it takes to reach its highest point is _____ Of the total round trip time

If you throw a rock upward at 64 m/s. How long does it take to reach its highest point? How long after it hits the highest point does it take to reach your hand again? Dx ofv equations = a= Dt Dv Dt Total round trip time? (v 1 + v 2) ofv equations = 2 How high did it go? y 2 = y 1 + v 1 t + at 2 2 What was its velocity when it hit your hand again? How fast is it going if it lands 35 meters below your hand? v 22 = v 21 + 2 a(y 2 - y 1)

If you throw a rock upward at 29 m/s. At what time is it 2. 0 meters above your hand? Dy= v 1 t + at 2 2 2. 0 m = 29 m/s t + -4. 9 Rearrange Dx ofv equations = a= Dt Dv Dt (v 1 + v 2) ofv equations = 2 2 2 y = y + v t + at 2 2 1 1 m/s t 2 v 22 = v 21 + 2 a(y 2 - y 1) 0 = -4. 9 t 2 + 29 t - 2. 0

0 = -4. 9 m/s t 2 + 29 m/s t -2 m 2 0 = a x 2 + b x +c x= t =. 069 sec, and 5. 8 sec Which one is right?

. 069 sec 5. 8 sec 2 m Time not to scale but you get the point?

Key things to remember about throwing upward problems the time to go UP is the SAME as the time going down IF…. . so the round trip is twice the time it takes to reach v=0 the speed at which it hits the ground is the same as the speed that it left (is the velocity? )

Acceleration due to gravity problems: Honors Physics: Page 44 Problems 34 – 38, 41, 44, 46 34 60. 0 m 35 8. 81 s, 86. 3 m/s down 36 32 m, 5. 1 s 37 1. 5 s 38 13 m 41 5. 22 s 44 +/- 12. 8 m/s, 0. 735 s, 3. 35 s 46 1. 8 m

Few review questions

A slingshot launches a marble into the air, and it hits the ground 5. 2 s later. Find: Time ascending Time descending highest position initial velocity

A bullet is fired straight up into the air at 440 m/s. When does it hit the ground?

An astronaut drops a hammer from a height of 1. 6 m, and it hits the aliens planet’s surface 2. 5 seconds later. What is the acceleration due to gravity on this planet.

A rock is dropped from a building, After 3 seconds, another identical rock is dropped from the same spot. What happens to the distance between the rocks while both are in free fall? ?

Few review questions

A slingshot launches a marble into the air, and it hits the ground 5. 2 s later. Find: Time ascending Time descending highest position initial velocity