Ch_4BranchiniB

**Chapter 4, Section 1**
toc
 * What do you think?**
 * The part that produces the loudest screams is when there is the first drop involved and when the coaster accelerates. The rider has no idea what to expect. These are the most fun and the fear of not knowing what's next is unbelievable.


 * Roller Coaster Internet**
 * King Da Ka - Six Flags Great Adventure, Jackson, NJ
 * This roller coaster has a maximum speed of 128 miles per hour and a maximum height of 456 feet. The rider goes from 0-128 in the span of 3 1/2 seconds. This is an insane roller coaster if you really want a rush.
 * Expedition Everest - Walt Disney World, Orlando, FL
 * This roller coaster has a maximum speed of 50 miles per hour. You also go backwards traveling at speeds of 35 mph. It has a drop of 80 ft, when you really feel the rush. You are getting chased by the yeti snowman throughout the ride.


 * Physics Talk**


 * Summary**
 * Scalar is a quantity that has magnitude, but no direction. Displacement is a measured distance with a direction included.
 * A vector is a quantity that has both magnitude and direction.
 * Displacement is the difference in positions, which it only depends on the endpoints. It does not follow a path.
 * Speed is the distance traveled divided by the elapsed time.
 * Velocity is the displacement divided by time where d elta is represented by final velocity minus initial velocity.
 * Acceleration is the change in velocity divided by time elapsed. It is a vector quantity. It is always in the direction of the change in velocity.

1. Distance is a scalar quantity, which means it has no direction. On the other hand, displacement is a vector quantity, which means it depends on direction. 2. Your displacement would be 0 because you went back where you started from. 3. Speed is a scalar quantity represented as distance/time, while velocity is a vector quantity represented as displacement/time. 4. You can find the acceleration of an object by the formula of A = Change in velocity (Vf - Vi) / Time elapsed.
 * Checking Up**


 * Physics To Go**

1. 2. In the Terminator Express, the biggest thrill will be from the highest points of acceleration around the back curve, the first drop in the beginning, and the horizontal loop at the end. 3. a) La Paz, Bolivia obtains the greatest speed. This is because the bigger the radius equates more distance. The formula is V = d/t. So, more distance means more velocity if the time is constant. b) V = d/t v = 4,000/24 = 1,667 km/hr c) The rider would not feel a thrill because there would be constant speed, which means no acceleration. 4. A = change in velocity/change in time a = 16 - 4/3 a= 4 m/s^2 5. a) Speed b) Velocity c) Acceleration d) Velocity and Displacement e) Displacement 6. V = d/t = .1/2 = .05 m/s 7. v = d/t .05 m/s = .05 m/time time = 1 second 8. A = change in velocity/change in time a = 25 m/s / 10 seconds a = 2.5 m/s^2 10. a) I would probably make the rides height (lift) a little shorter and less steeper. I would probably decrease the length of the drop and the speeds of the vehicle because the preschoolers would not be able to handle such thrills. Instead of a horizontal loop, I would add in a short drop at the end. b)


 * What do you think now?**
 * After reading this chapter, I can conclude that when the vehicle changes direction and during parts where the vehicle accelerates the most, as well as the biggest drops will produce the most thrills. All these turns will create a lot of screams. In conclusion, increasing speeds and accelerating will create the best thrill.

Chapter 4, Section 2

 * What do you think?**
 * The steel roller coaster would give you the biggest thrill. Because it is such a steep angle of 90, the steel roller coaster will have much more thrill. In order to obtain the biggest thrill, you want a very steep angle. In this case, the steel roller coaster has a steeper descent than the wooden roller coaster.


 * Physics Talk**


 * Summary**
 * There are energy transformations in roller coasters.
 * Gravitational Potential Energy is defined to be the resulting energy based on its position in a gravitational field.
 * Kinetic Energy is defined to be the energy an object possesses because of its motion, which is due to its speed.
 * GPE and KE both depend on mass.
 * The sum of GPE and KE is constant.
 * As GPE increases, KE will decrease.
 * Mechanical Energy is known as the sum of GPE and KE.
 * KE does not depend on direction.
 * A Joule is what one uses to measure energy.
 * [[image:sdffsadfe.png]]


 * Checking Up**

1. The length of the incline does not have any affect on the speed at the bottom. 2. When the height increases, the GPE will increase, as well. This goes the same for mass. When the GPE decreases, the mass will decrease. 3. As the velocity increases, the KE will increase also. KE will increase when the mass increases, too. 4. The GPE is converted to KE. The KE will increase because of the increase of velocity. 5. If you look at the chart above, you can see that there os 30,000 J of KE.


 * Physics to Go**

1. Although the inclines differ, both Track A and B will have the same end result of velocity because their initial height are equal. 3. 4. 8. The speed will not change. The GPE at the top is always equal to the KE at the bottom. 9. a) It is traveling the fastest at Point B. Due to the lift at point A, there will be an increase in velocity at point B (the bottom of the ramp). It is also at its lowest position which equates to a higher KE. b) Both points C and E is when the ride is traveling at the same speed. They have similar heights, thus concluding GPE = KE. c) The ride will travel faster at point D. There is more KE and less GPE.
 * Position of cars height (m) || GPE(J)=mgh || KE(J)=1/2mv^2 || KE+GPE(J) ||
 * Top (30) || 60000 || 60000 || 60000 ||
 * Bottom (0) || 0 || 60000 || 60000 ||
 * Halfway (15) || 30000 || 30000 || 60000 ||
 * 3/4 way down (7.5) || 15000 || 45000 || 60000 ||


 * What do you think now?**
 * The roller coaster with the highest maximum height will travel the fastest. In this case, the heights are both equal. Yet, Roller coaster with steel will have the greatest thrill because it has the greatest acceleration. This is due to the steeper slope.

Chapter 4, Section 3

 * What do you think?**
 * To get to the highest point, a roller coaster usually goes slowly and gradually reaches the top. No, it will to cost more to lift the people. The number of passengers is irrelevant.


 * Physics Talk**
 * Conservation of Energy: All energy in a system balances between KE, EPE, and GPE.
 * EPE is also SPE --> the energy stored in a spring due to its compression or stretch.
 * Bouncy balls lose their height because of the energy being converted into sound or heat energy.
 * The sum of energy are always the same.
 * The total number on the right should be equal (joules) throughout the whole way.
 * SPE is the same, yet a pop-up toy with more mass will not go as high as one with a lesser mass.
 * Larger mass equates to smaller height and less mass equates to higher height.
 * KE is converted into thermal energy, as cart breaks stop.
 * The formula for SPE: 1/2kx^2


 * Checking Up**

1. All of the SPE transfers into GPE and KE. 2. The toy will have 2 Joules of KE as well because it has 2 joules of SPE. 3. As the toy reaches maximum height, the GPE will equal 2 Joules. 4. SPE = 1/2kx^2. K is the spring constant and x is the measurement of compressed distance.


 * Physics to Go**

5. It will be higher than the first hill because the GPE is not enough to reach the top of the second hill. You can only get to the same initial height. 6. Roller coaster cannot travel forever because there is friction that stops this. There is work due to friction. This lowers the GPE and KE.

7. GPE = mgh GPE = (300) (9.8) (15) GPE = 44,100 J

8. a) KE = 1/2mv^2 KE = 1/2 mv^2 KE = 1/2 (400) (15)^2 = 45,000J b) GPE = KE = 45,000 J c) 45,000 = (400)(9.8)(h) Height = 11.5 meters

9. When the ball is thrown upwards, the height increases, this would mean that the GPE also increases. 10. It is the same because it is independent. They are all going up to the same height so they will all have the same gravitational potential energy at the end.

11. a) They should be the same. GPE = mgh .078 J b) KE = SPE 1/2 mv^2 = SPE SPE = 1/2 (0.020) (2.7)^2 SPE = 0.073 J c) KE = GPE 1/2mv^2 = mgh 1/2 (.006) (2.7)^2 = (.006) (9.8)h height = .13 meters

12. a) SPE = 1/2 kx^2 52,920J = 1/2k(4)^2 x = 6,615 Nm/s b) GPE = SPE GPE = mgh GPE = 70,560 J 1/2(6,615)x^2 = 70,560 k = 4.62m

13. KE = SPE KE = 1/2kx^2 KE = 1/2(40)(.3)^2 KE = 1.8 J


 * What do you think now?**
 * Today, roller coasters get up to their highest by the technology of cables or motors. When the roller coaster rises, the GPE shall rise with it. It will cost more because when you increase the mass, the maximum height will decrease. The cables need to be able to pull this particular mass. Using the Law of Conservation of energy, we can say that the energy is uses to obtain maximum height can equal the GPE when you are at the top of the hill.

**Chapter 4, Section 4**

 * What do you think?**
 * Gravity has a direction that points downwards. Gravity will always pull towards to the center of the earth. This is why anyone from Australia can stand.


 * Physics Talk**
 * Gravitational field: the gravitational influence in the space around a massive object
 * Earth is a source of gravitational field
 * Inverse square relationship: the relationship between the magnitude of a gravitational force and the distance from the mass. This also describes how electrostatic forces depend on the distance from an electrical charge. For example, when you triple the distance, you are decreasing the force by 1/9.
 * Acceleration due to gravity is greater when closer to the Earth.
 * Gravity: the force of attraction between two bodies due to their masses
 * Newton's law of universal gravitation: All bodies with mass attract all other bodies with mass; the force is proportional to the product of the two masses and gets stronger as either mass gets larger; the force decreases as the square of the distances between the two bodies increase.


 * Checking Up**

1. The direction of a gravitational field is towards the ground, pointing downwards. 2. The field lines are strongest when at the bottom near the Earth's core. 3. When tripling the distance between the masses, the force ends up being 1/9 of the initial. 4. The force that holds the moon in its orbit is gravity. 5. The shape of the orbit of the plants around the sun is elliptical (circular).


 * Physics to Go**

1. If you were to double the gravitational force between two asteroids, the force would 125 N because it is 1/4 of the initial force. 2. a) Force g = 1/4 of the initial (For 2, the relationship is an inverse squared) b) Force g = 1/9 of the initial c) Force g = 1/16 of the initial 3. Gravity is trusted because it is the force that keeps us standing. Due to gravity, nothing can float in the air. When you have mass, there is a gravitational force. 4. Acceleration due to gravity is less when at the top. 5. a) Water on the Earth's side facing the moon will be closer than the middle of the earth. b) There are high tides present on the bodies of water facing the moon. Gravitational fields are attracted to water, which makes the water rise. c) There is land all over the ocean, islands, thus meaning an unequal distribution of water. The bodies of water have different distances to the moon, which means they have weaker or stronger forces from the gravitational fields. 6. a) The fish would be swept up into the air and be a floating object, which would make it eventually die. b) Gravity holds the fish down due to the mass of the fish being much less than that of Earth's. The large mass of Earth is attracted to the small mass of the fish. 7. a) 1/4 of the force b) 1/9 of the force c) 1/16 of the force d) 4 times 8. a) 2 times b) 3 times c) 4 times d) 1/2 times 9. a) 4 times b) 9 times c) 16 times d) 1/4 times 10. a) 2 times b) 9 times c) 6 times


 * Physics Plus**

1. A c = v^2 / radius 2. V = d/t (2) (pi) (3.84) (10)^8 / 2440800 v = 998.5 m/s

3. A c = v^2 / radius A c = 998.5^2 / 3.84 x 10^8 A c = 0.0025 m/s ^2


 * What do you think now?**
 * Gravity definitely has direction. It is related to the direction of the particular mass. It will always point to the center of the earth, which happens to be the larger mass.
 * People in Australia are still standing, yet gravity should have them upside down.

**Chapter 4, Section 5**

 * What do you think?**
 * Yes, you can use the same scale because it can give a weight for both animals. A bathroom scale works by applying the force downwards (your body) and it applies a force upwards to obtain an accurate weight.


 * Physics Talk**
 * Hooke's Law: the restoring force exerted by a spring is directly proportional to the distance of the stretch or compression of the spring.
 * You use the formula: Fs = -kx
 * X is referred to the distance of compression and K is referring to the spring constant given.
 * The negative refers to the opposite direction of the spring constant.
 * When the spring is strong, there will be a higher spring constant value. When the spring is weak, there will be a lower spring constant value.
 * Weight will change due to the gravitational field and mass will always stay the same.
 * There is a spring inside a bathroom scale that compresses to obtain an equal force to equal the weight of your body.


 * Checking Up**

1. There is a direct relationship between the two. So, if this were to happen, the stretch distance would increase by 5, too. 2. The spring constant of a spring refers to the particular strength of the spring. When the K sum is large, there must be a stronger spring. 3. Mass is used as kilograms in the equation of weight. The formula, N = kg * m/s^2 shows that mass is directly related to weight. 4. When there is more weight applying down towards the scale, there will be a greater compression inside the scale.


 * Physics To Go**

1. a) w = mg = 100 * 9.8 = 980 N b) w = mg = 10 * 9.8 = 98 N c) w = mg = 60 * 9.8 = 588 N

2. a) .25/130 = 1/x x = 520 N b) .25/1000 = 1/x x = 4000 N c) .25/50 = 1/x x = 200 N

3. a) b) see the graph above c) The slope of the graph is 0.1491 N/cm. d) This is determined to be the spring force constant. e) The slope for the new graph has a higher slope. This equates to a stronger spring. Since it is a higher slope, the spring force constant is much bigger, making it a stiffer or stronger spring. 4. Fs = -kx 12 = .03k k = 400 N/m

5. Hooke means that when you increase the force of an object, the stretch will increase with it. They have a direct relationship with one another. 6. The spring constant of 15 N/m would require more effort to stretch because of its greater value of K. 7. K = the slope of the graph = 3/2 = 1.5 N/m

8. A spring scale shows how much weight a particular mass has in newtons. When you put the mass on the hook, the spring scale will stretch and you see what the weight is by looking at the scale.


 * What do you think now?**
 * You cannot weigh both a canary and an elephant on the same scale. Any scale works with the compression of a spring. The spring used for an elephant would have be much more stronger than a spring to weigh a canary due to the fact that the elephant's weight requires a higher spring constant.
 * A bathroom scale works by compression of a spring. The mass of your body applies a force to do this. The spring will compress any distance and it will apply the same force upwards.

**Chapter 4, Section 6**

 * What do you think?**
 * When you are riding a roller coaster, your weight will not change. The person riding may feel lighter or heavier during the roller coaster, though.
 * A bathroom scale when sat on, will give out different values based on where you were during the ride. It is obtaining the force that it is on the scale.


 * Physics Talk**
 * Newtons First Law: An object at rest will not have any net force acting upon it.
 * Newtons Second Law: An object at rest will have no acceleration and no force acting upon it.
 * When one is accelerating upwards, the force is upwards, too
 * When you sit on a scale on a roller coaster that is level, the scale will read a equal weight to that of the rider.
 * When traveling at constant speed, you have no net force nor acceleration.
 * Upward acceleration accounts for increase of weight, which accounts for an upward force.
 * Downward acceleration accounts for decrease of weight, which accounts for a downward foce.
 * Air resistance cannot be ignored on any roller coaster.
 * When an elevator is accelerating downwards, the reading on the scale is actually smaller of a force.

1. When you are moving at constant speed, the forces have a total net force of 0. 2. It will read a greater amount on the bathroom scale than the person's actual weight. 3. As one accelerates upwards, there is an upward force and these forces hold your stomach in place, so you feel heavier. 4. If the cable of an elevator would break, the forces would all point down. It would read a total of zero because it is a free fall. 5. The force is air resistance.
 * Checking Up**

1. a) vf = vi + at = (9.8) (2) = 19.6 m/s
 * Physics To Go**

b) vf = vi + at = (9.8) (5) = 49 m/s

c) vf = vi + at = (9.8) (10) = 98 m/s

2. a) vf = vi + at = (1.6) (2) = 3.2 m/s

b) vf = vi + at = (1.6) (5) = 7.5 m/s

c) vf = vi + at = (1.6) (10) = 16 m/s

4. 5. The elevator would be moving downwards with an increasing speed. This causes the student's apparent weight to fall due to the fact that acceleration and force are moving down. 6. The velocity and acceleration both point upwards. The net force always point the same direction as acceleration. Thus, the reading on the scale would be greater. 7. a) The net force and acceleration are pointing down so the bathroom scale reading would be lesser. b) F = ma w - N = ma (50) (9.8) - N = (50) (1.5) N = 415 N
 * **Motion of the Elevator** || **Acceleration (up, down, zero)** ||  || **Relative Scale Reading (greater, less or equal to weight)** ||
 * At rest, bottom floor || zero ||  || equal ||
 * Starting at Rest, Increasing Up || up ||  || greater ||
 * Continuing to move, Constant Up || zero ||  || equal ||
 * Slowing down to top floor, Decreasing Up || down ||  || less ||
 * At rest, top floor || zero ||  || equal ||
 * Starting at rest, Increasing Down || down ||  || less ||
 * Continuing to move, Constant Down || zero ||  || equal ||
 * Coming to a stop on the ground floor || up ||  || greater ||

8. a) The scale reading would remain the same because it is not moving. It would remain 50 kg. w = mg = (50) (9.8) = 490 N

b) F = ma N - w = ma N - 490 N = 100 N N (force of scale) = 590 N

c) The reading would stay at 490 N. This is because all the forces are equal, as N = w. 9. In the picture on the left, the elevator is at rest or constant velocity. The scale would read the same as the person because there is no net force. In the picture in the middle, the bathroom would read 0 because there is nobody on the scale. If the person is in free fall, there should be no force having him stay down. In the picture on the right, the reading on the scale should be greater because there is upward acceleration and when you accelerate upwards, the force is greater. 10. Our roller coaster appeals to beginner roller coaster riders. It is for the first riders. Our roller coaster was some thrill with less fear, thus the acceleration would not be as quick as a larger roller coaster. The roller coaster will hit its highest points of acceleration during the loop and the steepest points downwards.


 * What do you think now?**
 * One's actual weight will not change when riding on a roller coaster. Yet, your apparent weight will change. This is because you feel as if your total weight is changing due to the forces. When you are traveling uphill, you feel heavier because the force is upwards. When you are traveling downhill, you feel lighter because the force is downwards. Thus, the scale would give different readings. Your weight would read less when the force makes you feel lighter and it would weigh more if the force made you feel heavier.

Chapter 4, Section 7

 * What do you think?**
 * When the coaster goes upside down, we do not fall because the gravity is keeping you on the ramp. The total force and acceleration are both pointing down towards the center.


 * Physics Talk**
 * Normal force: the force acting perpendicular to the surface
 * Centripetal force: any force directed toward the center that causes an object to follow a circular path at constant speed
 * Diagrams: [[image:ahdfdsfasdfee.png width="562" height="196"]]
 * As you can tell, the centripetal force required is more when at the bottom of a loop than at the top of the loop.
 * [[image:asdhfdsfaheyuge.png width="223" height="551"]]
 * The components of the equation for centripetal force is mass, velocity and the length of the radius
 * An object accelerates when there is a net force.
 * Due to the force being towards the center, the acceleration is directed toward the center.
 * Centripetal acceleration: the acceleration directed toward the center of a circle experienced by an object traveling in a circular path at constant speed.

4. The normal force is responsible for ones apparent weight on a coaster. 5. Centripetal force is always greater when you increase the mass, increase the speed, and decrease the length of the radius. Mass and velocity have a direct relationship with Fc, while the length of the radius is inversely related to the Fc.
 * Checking Up**
 * 1. Centripetal force makes an object travel in a circle. **
 * 2. Yes, this is known as centripetal acceleration when you travel at constant speed in a circle. **
 * 3. At the top of a loop, normal force and gravitational force are the two forces that provide the centripetal force. **

1. a) The car would be traveling in a circle. b) If the string broke, the car would then travel straight tangent to the circle. 2. a) Friction replaces the string of the toy car. b) If the tire were to hit a section of ice, the car would follow a path tangent to the circle. 6. a) The speed of the coaster did not change and remained at 20 m/s. b) The velocity of the coaster did change due to a change in direction.
 * Physics To Go**

c) 20^2 + 20^2 = c^2 c = 28.2 m/s tan-1 (x) = 20/20 x = 45 degrees 28.2 m/s at 45 degrees NW

7. Ac = v^2 / r = (20)^2 / (200) = 2 m/s^2

10. Fast-moving Roller Coaster Slow-moving roller coaster 13. a) heavier  b) not sure  c) heavier  d) heavier  e) you would feel normal  14. a) up  b) down  c) up  d) up  e) zero  f) towards the center  g) towards the center
 * || Required Centripetal Force || Force of gravity || Normal force ||
 * the top of the loop || 4000 N || 500 N || 3500 N ||
 * the bottom of the loop || 6000 N || 500 N || 6500 N ||
 * || Required Centripetal Force || Force of gravity || Normal force ||
 * the top of the loop || 800 N || 500 N || 300 N ||
 * the bottom of the loop || 2800 N || 500 N || 3300 N ||

1. a) The net force would increases. When the mass is increased, the velocity remains the same, the radius remains the same. Ultimately, this will cause an increase of the net force. b) When velocity increases, the net force increases too because the relationship between velocity and net force is a squared relationship. This means that if you double the velocity, you quadruple the net force. 2. The strength of the track, which is the force of the track, will be quadrupled because force and velocity have a squared relationship. 3. If the length of the radius increased, the net force will decrease because they are indirectly proportional to one another. 4. The larger the radius for the curve, the smaller the force required to keep the car moving along the curve. If the curve is tight, meaning a smaller radius, then a larger force is required. 5. If the student let go of the stopper, the cart would continue moving in a straight line tangent to the circular path.
 * Physics Plus**


 * What do you think now?**
 * When you go upside down in a roller coaster car, you will not fall out. This is because centripetal force keeps the passenger in a circular motion to have him remain seated throughout the loop. The acceleration is pointing towards the center, which means the centripetal force points towards the center, as well.

Chapter 4, Section 8

 * What do you think?**
 * It takes more energy to pull a cart up a steeper incline because it has a greater height.
 * It is more difficult to walk up a steep incline because there is more force required to finish the walk.


 * Physics Talk**
 * Work: the product of displacement and the force in direction; the energy transferred to an object
 * Displacement is parallel to the force
 * Along an incline, the product of the force and the distance always remains the same
 * When going up an incline, the work used is transferred into GPE
 * Springs or brakes are used to stop a roller coaster
 * When stopping a cart, the kinetic energy performed is transferred into spring potential energy
 * Power: the work done divided by the time elapsed; the speed at which the work is done and energy is transferred
 * Power is a measure of watts.
 * Watt: the SI unit for power; 1 W = 1 J/s

1. When the cart is at rest at the top, the energy is transferred to GPE. 2. The roller coaster gets its GPE from all the work done when traveling up the hill. 3. Truckers use a ramp because it requires less force, but still gets the same job done. 4. The coaster's kinetic energy is then converted into work due to the friction that takes place. 5. A watt is the SI unit for power.
 * Checking Up**

1. a) At the bottom, the GPE equals zero and at the top, the GPE is at its maximum value. b) The GPE changes into work. c) The spring energy is equal to work and GPE. d) EPE = 1/2kx^2 e) Just before it hits the spring, the total energy is a lot of GPE with some KE. f) It decelerates after it first hits the spring. 2. a) Work = 0 because force and distance are perpendicular to one another. b) W = F x d = (60) (0.5) = 30 J
 * Physics To Go**

c) W = F x d = (75) (40) = 3000 J

d) W = F x d = (500) (.7) = 350 J

3. A good way to do this is to tell people to stop wasting energy when it is not needed or useful. 4. The mass of the cart would increase. This would lead to more work, force, and GPE. 5. a) W = F x d = (10000) (20) = 200000 J

b) P = W / T = (200000) / (150) = 1333.3 W

6. There is work when trying to bring the cart up a hill. The GPE increases. There is all GPE when at the top. When you travel downwards, that same GPE converts into KE. When you are at the bottom of the hill, you have KE. When at the top of a vertical loop, you have KE and GPE. When going in a back curve, you have KE and a little bit of GPE. During a horizontal loop, you have KE. When you are breaking at the ramp, there is work being done due to friction.


 * What do you think now?**
 * You exert much more energy when pulling the roller coaster up a steep incline than a gentle incline. This is because the height is of a greater distance. W = GPE. The equation of GPE = mgh.
 * It is less difficult to walk up a gentle incline due to the fact that you need much more force for a shorter distance. There is less force for a longer distance. In this particular scenario, force and distance are inversely related.

Chapter 4, Section 9

 * What do you think?**
 * The parts of The Snake that are most thrilling include where there is the most acceleration. Also, it would be whenever there is a change in direction, thus a change in velocity.
 * It will still be fun because there are changes in direction.


 * Physics Talk**
 * Vector: a quantity with both a number and direction
 * Scalar: a quantity with no direction, but with magnitude
 * When determining a vector, you must take into account the displacement.
 * For adding or subtracting vectors, when you have two different directions such as north and east, you must make a triangle and use the Pythagorean theorem to figure out the hypotenuse because when dealing with vectors, you must deal with magnitude and direction.
 * An example of a scalar quantity is energy.
 * An example of a vector quantity is force.
 * Force of gravity is always downwards, while the normal force is always perpendicular to the track.
 * Energy and Force are related.
 * Weight does the work on the coaster and this increases the kinetic energy.
 * Changes in energy require a work being done by a force.

1. The Pythagorean theorem is needed when you add vector quantities. 2. Energy is scalar, while force is vector. 3. KE + GPE will always be equal, gravity and mass are constant, thus the GPE depends on the height, and same height equates to same speed. 4. Energy considerations do not depend on the path. Energy will always be the same during the whole entire ride. 5. Work is required. To determine work, use the equation W = applied force x distance.
 * Checking Up**

tan-1(5/5) = 45 degrees
 * Physics To Go**
 * 1. a) a^2 + b^2 = c^2 **
 * c = 7.07 m/s **

b) a^2 + b^2 = c^2 5^2 + 5^2 = c^2 c = 7.07 m/s at 45 degrees SW

2. This is because the heights are the same. All the GPE is at the top. This means that you have the same KE, as well as velocity. 3. a) scalar b) vector c) scalar d) vector e) vector f) vector g) scalar h) scalar i) scalar 4. a) scalar b) vector c) scalar d) vector 5. When looking at the ride considering energy, it uses KE to get up the first hill, as well as work. When at the highest point, you are at maximum GPE and all the energy from the work and KE are converted into GPE when at the top. When you travel down the hill, all this energy is now KE. When looking at the ride considering force, when the cart goes up the hill, there is an increase in weight that is holding the cart down and the normal force decreases. 6. a) b) c) It is easier to measure the force of coaster due to the fact that it is a straight track. The normal forces are perpendicular to the track, so it is much easier. 7. a) b) The GPE at A, B, and C are equal because they all have the same height. Thus, the coaster has the same amount of energy at those points. c) At the points A, B, and C, the kinetic energy is equal because their GPE is equal as well; due to the height. d) KE + GPE always equates to the same amount energy.


 * What do you think now?**
 * The ride will be most thrilling when the snakes go in different directions, left to right. This is because the change of direction changes the velocity, as well.
 * The ride will still be fun even if it is the same speed throughout. The velocity will always be changing because there will be changes in direction.

**Chapter 4, Section 10**

 * What do you think?**
 * The knowledge of how a ride can hurt or cause die can affect the thrill negatively.
 * If I found out that 1/2 of all roller coaster rides caused death, I would not go on any of them because it would be risking my life.


 * Physics Talk**
 * In a roller coaster ride, the maximum amount of acceleration in g's that is safe is 4 g's.
 * The equation for centripetal acceleration is Ac = v^2 / radius of loop
 * If you decrease the speed of the roller coaster, you will be decreasing the acceleration
 * If you increase the radius of the loop, you will be decreasing the acceleration
 * When a determining the amount of g's in a roller coaster, you use the acceleration of that particular point and divide that acceleration by 9.8
 * Wherever the centripetal acceleration is greater, the force is greater
 * Net force (equation for net force: Net Force = mv^2 / r) is equal to the total normal force and the weight
 * When at the top of a loop, acceleration should be greater than 9.8
 * You will feel much more heavy when the number of g's increase due to the increase of normal force upon you

1. The maximum safe acceleration is 4 g's. 2. Two ways to do this include decreasing the height, as the GPE will decrease, or making the radius larger. 3. The greatest acceleration is at the bottom of a loop. 4. The greatest normal force is at the bottom of a loop.
 * Checking Up**

1. To make the roller coaster safe, I would check the values of the acceleration in g's and make sure that at any point, the acceleration does not exceed 4 g's. The cart should also have enough acceleration when traveling at the top of a loop. 2. a) GPE = KE mgh = 1/2mv^2 (9.8) (h) = 1/2 (20) ^2 h = 20 m
 * Physics To Go**

b) A = v^2 / r = 20^2 / 12 = 33.3 m/s^2

c) 33.3 m/s^2 / 9.8 = 3.4 g's. This would be a safe coaster because it does not exceed 4 g's. d) (9.8) (4) = 39.2 m/s^2 39.2 = v^2 / 12 v = 21.7 m/s

e) 39.2 = v^2 / 7 v = 16.6 m/s

3. a) Ac = v^2 / r Ac = 25^2 / 10 Ac = 62.5 m/s^2

b) A / 9.8 62.5 / 9.8 = 6.4 g's -- this means that this roller coaster is not safe because it exceeds 4 g's.

4. a) GPE = KE mgh = 1/2mv^2 (9.8) (50) = 1/2v^2 v = 31.3 m/s

b) Ac = v^2 / r = 31.3^2 / 10 = 96 m/s^2

c) Ac = v^2 / r 96 = v^2 / 10 v = 24 m/s

d) Ac = v^2 / r Ac = 24^2 / 10 Ac = 58 m/s^2 e) The accelerations are safe because at the bottom or top of the loop, the accelerations do not exceed 4 g's.

5. a) KE = GPE mgh = 1/2mv^2 (9.8) (16) = 1/2v^2 v = 8.9 m/s

b) GPE = KE mgh = 1/2mv^2 (9.8) h = 1/2 (8.9)^2 h = 20 m

6. a) A = v^2 / r = 12^2 / 18 = 8 m/s^2

b) Fc = mv^2 / r = (900) (12)^2 / 18 = 7, 200 N c) The centripetal force is equated by the weight of the track and the normal force.

7. a) A = v^2 / r = 20^2 / 15 = 26.7 m/s^2

b) Fc = mv^2 / r = (900) (20)^2 / 15 = 24, 000 N c) The coaster is safe because the force exerted by the cart is 24,000, which is less than the maximum force of the ride, 25,000. 8. a) It won't change. This is because mass is not a factor of centripetal acceleration. b) It will be the same speed because mass will not affect it. c) It will need a stronger material. This is because the weight is increasing, which means that the normal force must be bigger.


 * Physics Plus**


 * 1. a) GPE = KE **
 * mgh = 1/2mv^2 **
 * 9.8(h) = 1/2(30)^2 **
 * h = 45.9 m **


 * b) Ac = v^2 / r **
 * = (30)^2/9 **
 * = 100 m/s^2 **


 * c) 100/9.8 = 10g **
 * This means that the rollercoaster is not safe. This is because it exceeds 4 g's. **


 * d) 4 g's = a / 9.8 = 39.2 m/s^2 **


 * Ac = v^2 / r **
 * 39.2 = v^2 / 9 **
 * 18.8 m/s = v **


 * e) Ac = v^2 / r **
 * 39.2 = v^2 / 7 **
 * 16.6 m/s = v **


 * f) GPE = GPE + KE **
 * mgh = mgh + 1/2mv^2 **
 * (9.8) (45.9) = (9.8) (18) + 1/2(v)^2 **
 * v = 23.4 m/s **


 * g) Ac = v^2 / r **
 * Ac = 23.4^2 / 9 **
 * Ac = 60.8 m/s^2 **


 * h) G's = acceleration / 9.8 **
 * 60 / 9.8 = 6 g's **
 * This means that the rollercoaster is not safe. This is because it exceeds 4 g's. **


 * i) N = 0 **
 * mg = mac **
 * 9.8 = v^2 / r **
 * 9.8 = v^2 / 9 **
 * v = 9.4 m/s **

2. n-w = ma n = w w = mg w = (9.8) (1000) = 9800 N n = 9800 N

3. a) 1st bullet: Fc points down; 2nd bullet: Fc points to the right; 3rd bullet: Fc points to the right b) 1st bullet: normal pushes down; 2nd bullet: friction force; 3rd bullet: Normal pushes you

4. Fc = mv^2 / r = (1000) (15)^2 / 12 = 18750 N

5. It would stay the same. It would be 18750 N.

6. a) same as number 4: 18750 N bcd) N - w = mv^2 / r n - 9800 = 18750 n = 28550 N


 * What do you think now?**
 * If someone were to get hurt or potentially die, the thrill of the roller-coaster would definitely be negatively affected. There would be less riders on the roller coaster due to the lack of safety. Also, I would definitely not risk my own life to go on a roller coaster where 1/2 the riders died. When on a roller coaster, the acceleration cannot exceed 4 g's in order to be safe. If one is not safe, you can decrease the speed, height or increase the radius.