PHYSICS 30: Kinematics Constant Speed/Average Speed Practice

Physics 30 – Kinematics Unit VOCABULARY:  distance  displacement  speed  velocity  acceleration  tangent  relative velocity

SKILLS:  graph displacement vs time, velocity vs time, acceleration vs time  calculate slope  calculate area “under’ the graph  develop equations from slope, area, or y=mx+b  use kinematics equations to solve one dimensional motion problems  use kinematics equation to solve relative motion problems

of 23 Physics 30 – Kinematics Unit

PHYSICS 30: Kinematics – Constant Speed Activity

PURPOSE:  To create graphs describing constant speed motion.  To develop understanding of the meaning of slopes and areas from different types of graphs.  To develop general equations to describe constant speed motion.

THEORY:  The ticker timer creates dots at a rate of 60 dots per second. 12 dots = 0.2 secs

HYPOTHESIS:  How will you know if the dots on your paper tape are showing a constant speed?

 Sketch what you believe each of the following graphs would look like for an object traveling at a constant speed. Explain your graphs.

Distance Velocity

Time Time

APPARATUS & PROCEDURE: Sketch the apparatus. Label the apparatus. Explain how the apparatus was used.

DATA: Interval Time t Interval displacement Total distance Velocity v (cm/s) (s) d (cm) d (cm) v = d/0.2 s (interval displacement/ 0.2 s) 0 0 0 0 *** 1 0.2 2.70 2.7 13.5 2 0.4 2.65 5.35 13.25 **Interval displacement = length of 12 dots

GRAPHS: Create graphs of:  Total Distance versus Time: include a “trend line” or “best fit” line for the points you have plotted  Velocity versus Time: include a “trend line” for the points you have plotted. Draw this as a horizontal line. Page 3 of 23 Physics 30 – Kinematics Unit ANALYSIS:

Distance versus Time Graph: 1. Slope: a) Calculate the slope of the line you have drawn. Be sure to include the units of the slope. b) What is the meaning of the slope of the line you have drawn? The units are a good hint. 2. y-intercept: a) What is the value of the y-intercept from your graph? Include the units. b) What is the meaning of the y-intercept from your graph? 3. Equation: a) Use y=mx+b to develop an equation for your line. Don’t use y and x as variables in your equation, use the variables identified on your graph. b) Use the equation for your line in part a) to develop a general equation for constant speed motion. No numbers, just variables. Define any new variable you are using.

Velocity versus Time Graph: 4. y-intercept: a) What is the value of the y-intercept? Be sure to include the units. b) What is the meaning of the y-intercept?

5. Area: Draw a vertical line at some time from the line of v = 0 up to the straight line on the graph. a) Calculate the area of the region created. Be sure to include units in your calculation. b) What is the meaning of the area calculated? c) Write a general equation to describe the meaning of the area from a velocity versus time graph.

CONCLUSION:  R: Recall what you did to create your data for the activity  E: Explain the purpose of the activity  R: Results o Describe the shape of each graph created and why they have these shapes o What are the meanings of the slopes or areas of each graph? o What general equations did you develop? o How can you confirm the meanings of the slopes or areas?  U: Uncertainty o what are some systematic errors that might have occurred o do you have any results that you have questions about  N: New o What did you learn? o Do you have any questions? o What are applications to the real world for the work you did in the lab or its results?

of 23 Physics 30 – Kinematics Unit Physics Graphing Guidelines

Title Name: (Y Label vs X Label)) Partner: Date:

. Draw a circle around each data point that you plot. Use either an x or a point

Symbol

(units)

. Best Fit Line: o If the plotted points form a fairly straight line, then use a ruler and draw a straight line

o If the plotted points form some sort of a curved shape, then freehand draw a smooth curved line through the points

2 4 6 8 10 12 14 16 18 Scale example

Time t (s) Label example

of 23 Physics 30 – Kinematics Unit PHYSICS 30: Kinematics – Constant Speed/Average Speed Practice

d = v t km/h → m/s divide by 3.6 v = d/t m/s → km/h multiply by 3.6 t = d/v

1. The following chart shows world record times for some track events. Calculate the average speed. EVENT DISTANCE TIME AVERAGE SPEED MENS 100 M 100 m 9.77 s WOMENS 200 M 200 m 21.34 s MENS 400 M 400 m 43.18 s

2. Saskatoon is located 275 km from Regina. a. How long would it take to drive to Saskatoon at a speed of 110 km/h?

b. How fast would a plane need to fly in order to arrive in Saskatoon in 1.25 hours?

3. Winnipeg is located 600 km from Regina. a. How fast would you need to drive in order to arrive in Winnipeg in 4.5 hours?

b. How long would it take to reach Winnipeg flying at a speed of 350 km/h?

4. It takes 1 hour and 15 minutes to fly from Edmonton to Regina. a. Change the time to hours.

b. Calculate the distance from Edmonton to Regina if a plane flies at a speed of 675 km/h.

5. A Funnycar travels 400 metres in a time of 7.63 seconds. The Funnycar accelerates for the whole 400 metre track and reaches a speed of 377 km/h at the finish line. a. Calculate the average speed in m/s.

b. Change 377 km/h to m/s.

c. Compare your answer for a) and b). Why are the values different?

of 23 Physics 30 – Kinematics Unit Relative Motion: when objects are moving towards or away from each other they have relative motion.  The relative velocity of two objects each to each other is the difference of their velocities

Car A is chasing car B. Car A is travelling with a velocity of 20 m/s [E]. Car B is travelling with a velocity of 15 m/s [E].

The velocity of Car A ‘relative to’ Car B is:

From Car B, it appears that Car A is travelling with a velocity of .

The velocity of Car B ‘relative to’ Car A is:

From Car A, it appears that Car B is travelling with a velocity of .

Car A and Car B are moving towards each other. Car A has a velocity of . Car B has a velocity of . The velocity of Car A relative to Car B is:

From Car B, it appears that Car A is travelling with a velocity of .

of 23 Physics 30 – Kinematics Unit Example Questions:

1. A police car is chasing a red sports car. The police are travelling at a velocity of 40m/s [N]. The red sports car is travelling at a velocity of 32 m/s [N]. The police car is 700 metres behind the red sports car at the start of the chase. a. What is the relative velocity of the police to the red car?

b. What is the relative velocity of the red car to the police?

c. How long will it take the police to catch up to the red sports car?

2. A passenger train is travelling at a velocity of 25 m/s [W]. On the same set of tracks a freight train is travelling toward the passenger train with a velocity of 30 m/s[E].The two trains are 2700 metres apart. a. What is the relative velocity of the passenger train to the freight train?

b. What is the relative velocity of the freight train to the passenger train?

c. If the two trains continue on a constant velocity, how long will it be until they collide?

3. Two cars leave the same gas station at the same time. The Blue car travel west at 20 m/s. The Yellow car travels east at 25 m/s. a. What is the relative velocity of the Blue car to the Yellow car?

b. What is the relative velocity of the Yellow car to the Blue car?

c. How much time will it take for the two cars to be 5 km apart?

PHYSICS 30: Kinematics – Accelerated Motion Activity

PURPOSE:  To create graphs describing accelerated motion.  To develop understanding of the meaning of slopes and areas from different types o graphs.  To develop general equations to describe accelerated motion.

HYPOTHESIS: Sketch what you believe each of the following graphs would look like for accelerated motion.

Distance Velocity Acceleration

Time Time Time

APPARATUS & PROCEDURE: Sketch the apparatus. Label the apparatus. Explain how the apparatus was used.  12 dots = 0.2 seconds

DATA: A B C D E

1 TIME INTERVAL DISTANCE TOTAL DISTANCE SPEED (cm/s) ACCELERATION (s) (cm) (cm) (cm/s/s) 2 0 0 0 - -

3 0.2 <12 dot distance> <12 dot distance> <(Interval distance) / 0.2 s = <∆v/0.2 s> B3/0.2 > ** no value ** 4 0.4 <12 dot distance> <24 dot distance = C3 + <=(D4-D3)/0.2> B4>

GRAPHS: Create and print graphs of:  Total Distance versus Time – xy scatter with smooth lines  Velocity versus Time – xy scatter with best fit  Acceleration versus Time – xy scatter with best fit Include a “best-fit” line (trend line) for each graph. Print your data table as well as the graphs.

of 23 Physics 30 – Kinematics Unit ANALYSIS: Total Distance versus Time Graph: 1. Draw a tangent to the curved line somewhere in the middle of the curve. (Tangent: straight line that just “touches” the curved line at one point) a) Calculate the slope of the tangent line you have drawn. Be sure to include the units of the slope. b) What is the meaning of the slope of the tangent line you have drawn? (Check your units!)

Velocity versus Time Graph: 2. Slope: a) Calculate the slope of the line on the graph. Be sure to include the units of the slope. b) What is the meaning of the slope?

3. y-intercept: a) What is the value of the y-intercept? Be sure to include the units. b) What is the meaning of the y-intercept?

4. Equation: a) Use y = mx + b to write an equation for your line. b) Write a general equation for the equation of straight lines on graphs of velocity versus time.

5. Area: Draw a vertical line at some time from the line of v = 0 up to the straight line on the graph. a) Calculate the area of the region created. It may be easier if you divide the region up into rectangles and triangles. Be sure to include units in your calculation. b) What is the meaning of the area calculated? c) Write a general equation to describe the meaning of the area from a velocity versus time graph.

Acceleration versus Time Graph: 6. Area: Draw a vertical line at some time from the line of v = 0 up to the straight line on the graph. a) Calculate the area of the rectangular region created. Be sure to include the units. b) What is the meaning of the area calculated? c) Write a general equation to describe the meaning of the area from an acceleration versus time graph.

of 23 Physics 30 – Kinematics Unit Checking Meanings: 7. Total Distance vs Time Graph: a) The slope of the tangent line corresponds to a particular time. Check your data table for the velocity value at the same time. How does the slope value compare to the velocity value?

8. Velocity vs Time Graph: a) Compare the slope from the Velocity vs Time graph with the y-intercept from the Acceleration vs Time graph. b) The area of the region you created corresponds to a particular time. Check your data table for the total distance at the same time. How does the area compare to the total distance value?

9. Acceleration vs Time Graph: a) The area of the region you created corresponds to a particular time. Check your data table for the velocity value at the same time. How does the area compare to the velocity value? Check the change is velocity for the same time period.

CONCLUSION:  R: Recall what you did to create your data for the activity  E: Explain the purpose of the activity  R: Results o Describe the shape of each graph created and why they have these shapes o What are the meanings of the slopes or areas of each graph? o What general equations did you develop? o How can you confirm the meanings of the slopes or areas?  U: Uncertainty o what are some systematic errors that might have occurred o do you have any results that you have questions about  N: New o What did you learn? o Do you have any questions? o What are applications to the real world for the work you did in the lab or its results?

of 23 Physics 30 – Kinematics Unit PHYSICS 30: Kinematics – Equation Development

Displacement vs Time

Displacement d (m) Rise = ∆d

Run = ∆t

Time t (s) 1. Displacement vs Time graph; a. What does the slope represent on this graph? How do you know? b. Write a general equation for the slope of the tangent: slope = rise/run

Velocity vs Time

Velocity v (m/s) Rise = ∆v = vf - vi

vi Run = ∆t

∆t Time t (s) 2. Velocity vs Time graph: a. What does the slope of the line represent? How do you know? b. Write a general equation for the slope of the line: slope = rise/run c. Rectangular and triangular regions are created by the addition of a vertical line. What does the area of the rectangular and triangular regions represent? How do you know? d. Write a general equation for the calculations of the areas of the rectangle and triangle created by the vertical line. area = (L * W) + ½(B * H) e. What does the y-intercept of the graph represent? How do you know? f. Write a general equation for y = mx + b. Symbols only! Page 12 of 23 Physics 30 – Kinematics Unit Acceleration vs Time

Acceleration a (m/s2) a

∆t Time t (s)

3. Acceleration vs Time graph: a. A rectangular region is created by the addition of a vertical line. What does the area of this region represent? How do you know? b. Write a general equation for the area of this rectangular region: area = L * W

4. Compare the four equations you have created and search for similarities. What are the two common equations?

5. Equation Development through Substitution:

GRAPH ANALYSIS SUMMARY:  Distance vs Time Graph: o Slope →Velocity  Velocity vs Time Graph: o Slope →Acceleration o Area →Displacement (Distance)  Acceleration vs Time Graph: o Area →Velocity Page 13 of 23 Physics 30 – Kinematics Unit

PHYSICS 30: Kinematics – Graph Analysis

1. Velocity: a. How would you recognize constant velocity on this graph? b. During what times is the velocity constant? c. What are the values for the constant velocities? 2. Acceleration: a. How do you recognize constant acceleration on this graph?

b. During what times is the acceleration constant?

c. What are the values for the constant accelerations?

d. During what interval is the acceleration showing a consistent change?

e. How do you recognize this consistent change in acceleration on this graph?

f. Calculate the instantaneous acceleration at 15 seconds.

3. Displacement: a. How can you find the distance the object has traveled using this graph?

b. Calculate how far the object has traveled during the first 5 seconds.

c. Calculate the object’s displacement from 5 to 10 seconds.

Use a piece of graph paper to create graphs of:  Displacement vs Time  Acceleration vs Time

Use the same time scale for both.

PHYSICS 30: Kinematics – Equations

a = vf - vi t

2 d = vi t + ½ a t

2 d = vf t - ½ a t

2 2 2ad = vf – vi ÷ 3.6 km/h m/s d = ½ (vf + vi) t or d = vavg t x 3.6

Examples: 1. A bicycle starts at the top of a hill with a consistent slope. The bike is initially not moving and rolls down the hill which is 120 metres long. It takes the bike 9.75 seconds to reach the bottom of the hill. a. Calculate the acceleration of the bike as it travels down the hill. b. What is the velocity of the bike when it reaches the bottom of the hill?

2. A car traveling at a velocity of 50 km/h accelerates to a velocity of 80 km/h in 5.42 seconds. a. Calculate the acceleration of the car. b. What distance would the car travel while accelerating?

3. The acceleration of gravity is 9.8 m/s2 [Down]. A watermelon is dropped from the roof of a building 32 metres above the ground. a. How long does it take the watermelon to hit the ground? b. What is the watermelon’s velocity when it hits the ground?

4. A motorcycle accelerates from a stop-light at a rate of 5.4 m/s2 for 6.5 seconds. a. What distance does the motorcycle travel during its acceleration? b. What is the velocity of the motorcycle after 6.5 seconds of acceleration?

5. A car travels a distance of 500 metres in a time of 7.5 seconds. The speed of the acceleration car after traveling 500 metres is 200 km/h. a. What was the initial speed of the car at the start of the 500 metres? b. What was the acceleration of the car during the 500 metres?

6. A bicycle rider is traveling along a flat stretch of road and then begins traveling down a slope. The cyclist travels 300 metres down the slope reaching a speed of 72 km/h. The slope gives the bicycle a constant acceleration of 0.5 m/s2. a. How long did it take the cyclist to reach the bottom of the 300 metre slope? b. What was the initial speed of the cyclist at the beginning of the slope? Page 16 of 23 Physics 30 – Kinematics Unit

PHYSICS 30: Kinematics – Homework Assignment 1. A commuter train is traveling at a velocity of 90 km/h [E] brakes to a stop in 32.4 seconds. a. Calculate the train’s acceleration while braking. b. What is the direction of the acceleration? Explain how you know. c. Calculate the distance traveled by the train during braking?

2. A curler delivers her rock with an initial velocity of 2.5 m/s [N]. The rock experiences an acceleration of -0.11 m/s2 before coming to rest. a. What distance has the rock traveled when it stops? b. What is the meaning of the negative sign on the acceleration value? c. How long after the rock is released does it come to a stop?

3. Top-fuel drag racers can accelerate at a rate of 16.4 m/s2. A car accelerates from rest to a speed of 420 km/h when crossing the finish line. a. What is the length of the drag race course? b. How much time does it take to travel the length of the race course? c. If your reaction time off the start is 0.1 seconds slower than your competitor, how far in front of you would their car be when you begin moving?

4. A car traveling at a speed of 54 km/h begins accelerating at a rate of 1.8 m/s2 in order to increase its speed to highway speed. After accelerating for 222 metres the car merges onto the highway. a. How much time did it take the car to travel the 222 metres? b. How fast was the car traveling, in km/h, when it merged onto the highway?

5. An airplane must reach a minimum speed of 220 km/h in order to take-off from the runway. The runway at a particular airport is 1200 metres long. a. What is the minimum acceleration the plane must have in order to reach take-off speed? What assumption must you make in order to complete this calculation? b. How much time is needed for the plane to reach its take-off speed?

The pilot actually accelerates the plane at a rate of 4.2 m/s2 in order to take off before reaching the end of the runway: c. How much time is needed to reach take-off speed? d. What length of the runway is used to reach the take-off speed?

6. Most vehicles are capable of producing a maximum acceleration of -7.2 m/s2 when braking. The speed limit on the Lewvan Expressway is 80 km/h. There is a set of warning lights at the intersection of Lewvan and Regina Avenue that has been added for safety reasons. The warning lights begin flashing before the green traffic light turns yellow. The flashing light warns drivers that they need to begin braking as they will not pass through the intersection before the traffic light turns red. a. What is the minimum distance from the intersection that the lights can be located and still warn drivers (assuming maximum braking by the driver)? b. The warning lights are actually located 200 metres in front of the intersection. What acceleration was assumed by the traffic engineers when placing the warning lights? c. Drivers passing under the warning lights when they begin flashing continue on and pass through the intersection as the traffic lights turns red. How much time before the traffic light turns red should the warning lights begin flashing? Page 17 of 23 Physics 30 – Kinematics Unit 7. A motorist driving his car at a speed of 70 km/h suddenly sees the flashing yellow light of a construction barricade 40 metres ahead. It takes the driver 0.65 seconds to react and begin braking at a rate of -7.2 m/s2. a. What distance is traveled while reacting before braking begins? b. Does the car stop before reaching the barricade? Explain how you know this. c. If the car hits the barricade, what is its speed when it hits the barricade?

8. Police measure skid marks 74 metres long where a truck made an emergency stop. Police determine that at maximum braking the acceleration of the truck would be -8.4 m/s2. Was the truck exceeding the posted speed limit of 90 km/h? Justify your answer.

9. A car traveling at a speed of 140 km/h has just pulled beside a truck traveling at a speed of 100 km/h when both drivers spot a deer on the road ahead. Both drivers begin braking with an acceleration of -8.0 m/s2. a. What is the braking distance for the truck? b. What is the braking distance for the car? c. What is the truck’s braking time? d. What is the car’s braking time? e. How fast is the car still traveling when its braking displacement is the same as the truck’s braking distance?

10. A truck traveling at a speed of 70 km/h in a school zone passes a stationary police car. The police car begins accelerating at a rate of 3.2 m/s2 just as the speeding truck passes it. a. What do you know about the time and distance for the two vehicles when the police car catches up to the speeding truck? b. How is the motion of the truck different from the motion of the police car? c. How much time does it take for the police to catch the speeding truck? d. How far have the police traveled before catching up to the speeding truck? e. How far has the truck traveled? f. How does the police car’s speed compare to the truck’s speed when the police catch up to the truck?

11. In another high-speed chase incident the police begin chasing a motorcycle that is traveling at a speed of 145 km/h. The police car begins its chase just as the speeding motorcycle passes and the police car accelerates at a rate of 3.5 m/s2 until it reaches it maximum speed of 170 km/h. The motorcycle remains traveling at a speed of 145 km/h. a. How much time does it take the police to reach their maximum speed? b. What distance do the police travel in this time? c. Do the police catch the motorcycle before reaching their maximum speed? Explain. d. What distance does the police car travel before catching up to the motorcycle?

PHYSICS 30: Kinematics – Review

1. A bicycle rider is traveling at a speed of 8 km/h when she stops pedaling and begins accelerating down a hill at a constant rate of acceleration of 1.2 m/s2. The hill is 145 metres long. a) What is the cyclist’s speed at the bottom of the hill? b) How long does it take to travel from the top of the hill to the bottom? c) What is the average speed of the cyclist on the hill? d) How does the initial speed affect the time it takes to reach the bottom of the hill? Explain.

2. A ball is rolled up a slope with an initial velocity of 22 m/s up the slope. The ball has an acceleration of –2.4 m/s2 up the slope. a) What is the maximum distance the ball travels up the slope? b) At what two times is the ball at a position 30 metres up the slope? c) What are the magnitudes and directions of the ball’s velocities when it is at a position 30 metres up the slope?

3. A car is traveling at a constant velocity of 80 km/h [W] when the driver sees a deer ahead on the road. It takes the driver 0.75 seconds to react and begin braking. While braking the car has an acceleration of –7.5 m/s2 [W]. a) How far does the car travel from the time the driver sees the deer until he begins braking? b) What does the negative sign on the acceleration tell you about the car’s velocity? c) How could you write the acceleration as a positive value? d) What time is required for the car to come to a stop once braking begins? e) What total distance does the car travel from the time the driver sees the deer until the car comes to a stop?

4. A car is stopped on the side of the road. A van passes the car just as the car begins accelerating. The van is traveling at a constant speed of 105 km/h. The car keeps accelerating at a rate of 3.6 m/ss. a) How far does the car travel before it catches up to the van? b) How much time does it take the car to catch up to the van? c) What is the car’s speed when it catches up to the van?

5. A car and a truck are traveling on the highway at a constant speed of 90 km/h. The car is following the truck at a distance of 55 metres. The car pulls out to pass the truck and begins accelerating at a rate of 2.5 m/s2. a) What distance does the car travel in 5.0 seconds while attempting to pass the truck? Has the car passed the truck in 5.0 seconds? b) How much time does the car require to pass the truck? Use 55 metres as the distance the car needs to make up to pass the truck. c) What is the car’s speed when it finishes passing the truck if it maintains a constant acceleration? d) Explain why it is fairly important to be close to the lead vehicle when attempting to pass on the highway.

6. Answer these questions for the graph of Velocity versus Time. The graph describes the velocity of a car traveling on a highway. a) What is the magnitude and direction of the car’s velocity at 6 seconds? b) What is the magnitude and direction of the car’s acceleration at 6 seconds? c) What is the magnitude and direction of the car’s displacement from zero to 10 seconds? d) What is the magnitude and direction of the car’s acceleration at 25 seconds e) Describe what the driver is doing with the car during the 30 seconds of motion shown on the graph. f) When does the car back up? How do you know? g) When does the car have a changing acceleration? How do you know? h) When is the car traveling at a constant velocity? How do you know?

Car Velocity versus Time

10 ) ] N

s / m (

Series1 v

c 6 o l e V

0 0 5 10 15 20 25 30 35 40 Time t (s)

of 23 Physics 30 – Kinematics Unit Review Answers

1 a) vf = 18.8 m/s b) t = 13.8 s

c) vavg = 10.5 m/s d) the larger the initial speed, the less time it will take to reach the bottom of the hill and the larger the average speed will be 2 a) d = 100.8 m b) t = 1.5 s and t = 16.8 s

c) vf = ± 18.4 m/s 3 a) d = 16.7 m b) a negative acceleration means the velocity is decreasing c) -7.5 m/s2 [W] = +7.5 m/s2 [E] d) t = 2.96 s e) braking distance = 32.9 m → Total stopping distance: 16.7 m + 32.9 m = 49.6 m 4 a) d = 473 m b) t = 16.2 s

c) car’s vf = 58.4 m/s 5 a) d = 156 m → No car has not passed truck. Truck travels 125 m plus it was 55 m ahead of the car. Total = 180 m. b) t = 6.63 s

c) vf = 41.6 m/s = 150 km/h d) the less distance from the vehicle to be passed, the less time and distance is needed to pass the vehicle 6 a) v = 8.5 m/s [N] b) a = 0.75 m/s2 [N] c) c = 77.5 m [N] d) a = slope of tangent ≈ -0.13 m/s2 e) 0-12 s: speeding up (constant acceleration) 12-18 s: traveling at a constant speed 18-26 s: slowing down 26-35 s: travelling at a constant speed **** Never backs up, ALWAYS travelling forward f) Never backs up, velocity is always positive g) 18 – 26 seconds: graph is a curve, slope (acceleration is constantly changing) h) 12-18 seconds & 26-35 seconds: graph is a horizontal line, velocity is constant or slope (acceleration) is zero

of 23 Physics 30 – Kinematics Unit KINEMATICS REVIEW: http://www.physicsclassroom.com/reviews#1Dkin

Part A: Mulitple TRUE/FALSE 3 True: 4 True: 5 True: 6 True: Part B: Multiple Choice 8 9 10 11 12 acceleration= 13 displacement= average velocity= 14 15 17 19 20 24 25 28 Part C: Diagramming 29 a 29 b 29 c 29 d 29 e 30 Part D: Kinematics Graphing

position

of 23 Physics 30 – Kinematics Unit +

32 Velocity 0 Time

- 35 a 35 b 35 c 36 a 36 b 36 c 37 a 37 b 37 c Part E: Computational Problems 43 a 43 b 43 d 44 a 44 b 44 c 44 d 44 e 45 46 47 49 50