Chapter+Four

=Amanda Steiger's ActivePhysics Wikilogtoc= =Chapter 4=

Section One
Prompt: Which part of a roller coaster ride produces the loudest screams? Why? Response: Personally, I find the most frightening part of a roller coaster ride to be the moment before plummeting downwards. The apprehension and anxiety created by the knowledge of the "fall" awaiting the rider produces a surge of adrenaline, leading the riders to let out screams of excitement and fright.
 * What do you think?**

Prompt: Look up roller coaster design on the Internet and list at least two roller coasters mentioned. Describe their most important features. Response: The informative website learner.org discussed the evolution of roller coasters. The theme park Disneyland introduced perhaps the most well-known roller coaster, the Matterhorn, the first tubular steel coaster. Before the Matterhorn, roller coasters were composed of wood- a material that limited possible coaster designs. The website also addressed an unnamed Six Flags Magic Mountain roller coaster with a height of 415 feet and can reach 100 mph.
 * Homework 2/28/11**

The word scalar describes a quantity that has magnitude (size/amount) but no direction. A vector quantity has both magnitude and direction. Distance is a scalar quantity. Displacement, or change in position, is a vector quantity. Speed is defined as distance traveled divided by time elapsed. Speed is a scalar quality. Velocity is defined as displacement divided by time elapsed. Velocity is a vector quantity. Acceleration is the change in an object's velocity divided by time elapsed. Acceleration is a vector quantity.
 * Physics Talk: Measuring Velocity and Acceleration**

1. Distance is a scalar quantity, while displacement is a vector quantity. Displacement is a measured distance with direction included, and only depends on endpoints 2. In the situation described, displacement would be 0, because you start and end at home. 3. Speed is a scalar quantity, while velocity is a vector quantity. 4. Acceleration is found by dividing change in velocity by time elapsed.
 * Checking Up**

1. See notebook for sketch 2. Personally, I find the hilltop before the biggest drop to be the most exhilarating because of the apprehension. 3a. Bolivia is traveling faster because it travels a larger circumference in the same amount of time it takes Olso to travel a smaller circumference. 3b. If you are at Earth's equator, you are traveling approximately 1666 km/h. 3c. People are //always// traveling at such a high speed and are therefore are not "aware" or thrilled by the movement. There is no change in speed. 4. Acceleration is 4 meters per second squared. 5a. Speed 5b. Velocity 5c. Acceleration 5d. Displacement and velocity 5e. Displacement 6. 5 centimeters per second 7. 1 second 8. Acceleration is 2.5 meters per second squared. 10. If a roller coaster was to be designed for young students, two changes that should be made are smaller seats and restraints and lower hills.
 * Physics to Go**

Prompt: Research roller coasters on the Internet. Which are the most modern? What are some innovations in newer roller coasters? What features from historic coasters have been retained? Compare wooden and steel roller coasters. Response: The first roller coaster in the US was built in Coney Island in 1884, and traveled at approximately 8 miles per hour. Passengers had to push the train themselves so that it could return for new riders. A cable was built to solve this problem. Wooden roller coasters reigned until the introduction of the Matterhorn coaster at Disneyland, which was made of steel. This new building material gave way to "supercoasters" and allowed for bigger thrills. King Da Ka is perhaps the most modern roller coaster, boasting a 90 degree angle drop and the title "world's fastest and tallest roller coaster." Today, magnetic fields, turbines, and hydraulics are used to control coasters' movements.
 * Inquiring Further**

Prompt: Which part of a roller coaster ride produces the loudest screams? Why? Response: The parts of a roller coaster in which changes in direction or speed occurs produce the loudest screams, as they result a change in the rider's movements.
 * What do you think now?**

Section Two
Prompt: The steepest angle of descent on a wooden roller coaster is 70 degrees. The steepest angle of descent on a steel roller coaster is 90 degrees. Which roller coaster will give the bigger thrill? Why? Response: The 90 degree angle will give the bigger thrill as it produces a bigger change in direction and speed.
 * What do you think?**

Gravitational potential energy (GPE) is calculated by multiplying the mass of the object, the strength of the gravitational field, and the height above the ground. GPE is defined as the energy an object possesses as a result of its position in a gravitational field. Kinetic energy (KE) is calculated by multiplying 1/2, the object's mass, and the object's velocity squared. KE is defined as the energy an object possesses because of its speed. Energy is measured in joules (J). The term mechanical energy refers to sum of kinetic energy and potential energy. Total energy in a system must always remain the same.
 * Physics Talk: Gravitation Potential Energy and Kinetic Energy**

1. The length of the incline of a track does not determine the speed of a ball at the bottom of said track. 2. As an object's mass and/or height increase or decrease, the GPE of the object will increase or decrease, respectively. 3. As an object's mass and/or speed increase or decrease, the KE of the object will increase or decrease, respectively. 4. As a roller coaster cart rolls down a hill, the GPE it loses is converted into KE. 5. If a roller coaster car has 40,000 J of GPE when at rest on top of a hill, it will have 10,000 J of KE when it is 3/4 of the way down the hill.
 * Checking Up**

1. Carts A and B will have the same speed at the bottom of the hill because they were launched from the same height. 3. Mass of car = 200kg and g = 10 N/kg or 10 m/s squared (approximate value) 4. See notebook for sketch 8. The increase in mass will not affect the speed because mass "cancels" out. 9a. The roller coaster is traveling the fastest at point B (lowest point). 9b. The roller coaster is traveling at the same speed at points C and F because they have the same height. 9c. The roller coaster is traveling faster at D than it is at E because D has the lower position.
 * Physics To Go**
 * Position of Car --> Height (m) || GPE (J)=mgh || KE (J)=(1/2)mv^2 || GPE + KE (J) ||
 * Top (30 m) || 60000 || 0 || 60000 ||
 * Bottom (0 m) || 0 || 60000 || 60000 ||
 * Halfway down (15 m) || 30000 || 30000 || 60000 ||
 * Three quarters down (7.5m) || 15000 || 45000 || 60000 ||

Prompt: As a skateboarder practices on the vert (vertical surface), there are constant changes in the gravitational potential energy GPE and the kinetic energy KE. Research the size of the vert and report back on how the conservation of energy plays an integral part in this sport. You may also wish to make measurements of skateboarders in the vert from a video on the Internet. Response: Vert skateboarding is a term used to refer to riding a skateboard on a vertical ramp that is a larger version of a halfpipeIn the sport of skateboarding, energy is constantly being converted. The skateboarder goes up and down a U-shaped platform. The skateboarder descends from rest on the platform using KE which is converted into GPE when he or she soars to some height above the vertical surface.
 * Inquiring Further**

Section Three
Prompt: How does the roller coaster today get to its highest point? Does it cost more to lift the roller coaster if it is full of people? Response: Roller coasters today are controlled in several ways. It does cost more to lift a roller coaster if it is full of people- it will have an effect.
 * What do you think?**

The law of conservation of energy states that energy cannot be created or destroyed but may change form. Total energy in a system __never__ changes. At an object's highest point, the type of energy present is GPE (KE=0). Just after an object begins to move, KE is most prevalent. Elastic potential energy (spring potential energy) is calculated using the formula (1/2)(k)(x^2), where k is the spring constant and x is the amount of compression. EPE is defined as the energy stored in a spring due to its compression or stretch. The graph below depicts the amount of energy present at different locations in a spring toy's "journey."
 * Physics Talk: Conservation of Energy**

1. After a pop up toy leaves the table, its EPE decreases to 0. 2. After the pop up toy leaves the table, it will have approximately 2 J of KE. 3. When the pop up toy is at its maximum height, it will have 2 J of GPE. 4. The factors that affect the amount of EPE an object has are the spring constant (type of spring) and amount of compression or stretch.
 * Checking Up**

5. Total energy present in the first hill must be greater than the energy in the second hill. 6. Energy is lost due to friction and air resistance. The roller coaster cannot continue forever due to work! 7. GPE=mgh GPE=(300)(9.8)(15) GPE= 44100 J 8a. KE=(1/2)(m)(v^2) KE= 45000 J 8b. If KE is 0, GPE= 45000 J 8c. GPE=mgh 45000= (400)(9.8)(h) 11.48m= h 9. GPE is increasing 10. The same- they all have the same maximum height and equal mass 11a. Yes 11b. GPE=EPE mgh=EPE (0.020)(9.8)(0.4)=EPE 0.0784 J=EPE 11c. 0.0784=GPE 0.0784=mgh 0.0784=(0.06)(9.8)(h) 0.13m=h 12a. GPE=EPE 52920=(1/2)(k)(x^2) 6615 N/m= k 12b. GPE=EPE 70560=(1/2)(k)(x^2) 4.6m = x 13. EPE=KE (1/2)(k)(x^2)=KE 1.8 J = KE
 * Physics To Go**

Prompt: How does the roller coaster today get to its highest point? Does it cost more to lift the roller coaster if it is full of people? Response: A roller coaster is lifted up by cables and electricity. Changes in mass do affect the amount of energy needed to raise a roller coaster to its highest point. More electrical energy is required to lift a heavier roller coaster because more work is required. A heavier roller coaster means an increase in GPE, which means more energy is needed! The experiment we completed in class showed that the height of the pop-up toy decreased when the mass increased.
 * What do you think now?**

Section Four
Prompt: Does gravity have a direction? How can people in Australia be held on Earth when they are "upside down"? Response: Gravity is always directed towards the center of the Earth.
 * What do you think?**

A "field" is an influence that one object (Earth) sets up in the space around it. The first object is called the source of the field. Earth is the source of its gravitational field. A second object (your body, the moon, etc) interacts with this field. The second object is called the response or test object. The direction of the gravitational field is the direction of the force on a mass. Acceleration due to gravity is largest near Earth and gets weaker as you move further from Earth. Gravitational field is present everywhere- extends to infinity! Inverse-square relationship says that the force of gravity between two objects decreases by the square of the distance between them. An inverse-square relationship is defined as the relationship between the magnitude of a gravitational force and the distance between the mass... also describes how electrostatic forces depend on the distance from an electrical charge.
 * Physics Talk: Newton's Law of Universal Gravitation**

1. The direction of the gravitational field in the classroom is towards the center of the Earth. 2. Regarding field lines, the gravitational field is the strongest where the lines are closest together (closest to center of Earth). 3. If you triple the distance between two masses, the force of gravity between the two masses is divided by 9. (Inverse-square relationship) 4. The force that holds the Moon in its orbit around Earth is gravity. 5. The planets travel around the Sun in ellipses.
 * Checking Up**

The Law of Universal Gravitation
 * Class Notes 3/17/11**

Galileo (late 1500s) - Earth moves around Sun in circular path. Kepler (1600s) - Apprentice to Tycho Brahe... "Steals" data from Brahe and analyzes information

Kepler develops three laws 1. Law of Ellipses: Planets orbit Sun in an elliptical path and Sun is at one focus. The deviation of a curve or orbit from circularity is called eccentricity... e=0 in perfect circle. 2. Law of Equal Areas: The closer a planet is to the Sun, the faster it travels (and vice versa). Areas "carved out" by a planet in equal time intervals are equal. 3. Law of Harmonies: T squared (period- time it takes for planet to go around the sun) divided by R cubed (distance from center of planet to sun) is constant for all planets in solar system.

Newton (late 1600s) - Found that the force of gravity is inversely proportional to the distance between centers of 2 objects. All objects in the universe are gravitationally attracted to each other.

Cavendish (late 1700s) - G universal gravitational constant "big G" 6.67x10^-11 Nm^2/kg Fg= G x (m1 x m2)/d^2

Force of gravity= gravitational attraction = gravitational force = weight (N)

2a. The gravitational force would be 1/4 weaker. 2b. The gravitational force would be 1/9 weaker. 2c. The gravitational force would be 1/16 weaker. 3. Most people accept gravity because they are affected by it everyday. 4. The acceleration due to gravity must be weaker at the top of a roller coaster ride than at the bottom. The top of the ride is further from Earth’s center. 5a. The water on the side of Earth facing the Moon is closer to the Moon than Earth’s center. 5b. The Moon’s gravity “pulls” the water towards it. 5c. When water is bulging on one side, the other side must be depleted. 6. Without gravity, fish would die from lack of water.
 * Physics To Go**


 * Physics Plus**

Prompt: Does gravity have a direction? How can people in Australia be held on Earth when they are "upside down"? Response: Gravity is always directed towards the center of Earth. Because all objects are gravitationally attracted to each other, it is impossible for people in Australia to be unaffected by gravity. Australia is part of Earth's gravitational field. Additionally, the Earth is a sphere and Australia seems "upside down" to those in the US, but from another perspective, it may not be.
 * What do you think now?**

Section Five
Prompt: Can you use the same scale to weight a canary and an elephant? How does a bathroom scale work? Response: I think you can use the same scale to weigh both animals. I am not sure how a bathroom scale works.
 * What do you think?**

Hooke's Law states that the restoring force exerted by a spring is directly proportional to the distance of stretch or compression of the spring. The more you stretch a spring, the larger the restoring force of the spring. The equation for Hooke's law is force exerted by the spring = (-spring constant)(spring stretch) --> Fs= -kx. The spring constant k is an indication of how easy or difficult it is to stretch or compress a spring. A stiff spring will have a large k value, a "soft" spring will have a smaller k value. The spring constant k depends on the material from which the spring is made and the size and shape of the coils. The negative in front of kx informs us that the force is opposite the stretch/compression distance.
 * Physics Talk: Hooke's Law**

1. Five times 2. The spring constant is an indication of how easy or difficult it is to stretch or compress a spring. 3. The weight of an object in Newtons is found by multiplying the object's mass by the acceleration of gravity (9.8m/s^2) 4. The force of compression of the spring is equal to the force of your weight on the scale.
 * Checking Up**

1a. 980 N 1b. 98 N 1c. 588 N 2a. 520 N 2b. 4000 N 2c. 200 N 3. 3d. The slope represents the spring constant k. 4. k = 4 N/cm 5. Hooke's Law says that the force exerted by a spring is directly proportional to the stretch/compression distance of the spring. 6. The one with the greater k value (15 N/cm) is more difficult to stretch/compress. 7. The k value is 2/3 N/cm 8. When an object is placed on a spring scale, the spring must exert an equal force to that of the object on the scale. In this way, people are able to read the weight of an object by placing it on a spring scale.
 * Physics to Go**

Prompt: Can you use the same scale to weight a canary and an elephant? How does a bathroom scale work? Response: An elephant and a canary cannot be weighed on the same scale- they would need springs with different 'k' values. A bathroom scale works by employing a spring. The spring's compression should be equal to the weight of the object on the scale.
 * What do you think now?**

Section Six
Prompt: Does your weight change when you are riding on a roller coaster? If you were sitting on a bathroom scale, would the scale give different readings at different places on the roller coaster? Response: Your earthly weight will not change. However, the scale would provide varied weights because you are moving with the roller coaster and the scale (inertia).
 * What do you think now?**

Increasing velocity- acceleration and velocity are in same direction Constant or at rest**-** acceleration is equal to 0 Decreasing velocity- acceleration and velocity are in opposite directions
 * Class Notes 4/5/11**

Net force MUST be in the same direction as acceleration! (Fnet=ma) Bigger force always points in the direction of the net force.

As a roller coaster moves, its passengers experience "funny feelings" in their stomachs which can be explained by physics. When an object is at rest, acceleration is 0 and the sum of the forces (net force) acting on the object is 0 **(**balanced). This is explained by Newton's first law (inertia) and Newton's second law (Fnet=ma). The same is true of an object moving at constant speed- acceleration is 0**,** net force is 0. If a person was to sit on a scale in a level roller coaster cart at rest or moving with constant velocity, the scale reading would be accurate. The force of the Earth pulling on the person (his/her weight) would be equal to the force of the compressed spring within the bathroom scale. If the person was to sit on the scale while the cart accelerated upwards, the scale reading would be larger than the person's actual weight. The Earth is pulling the person down with less force than that being exerted by the spring. If the person was to sit on the scale while the cart accelerated downwards, the scale reading would be smaller than the person's actual weight. The Earth is pulling the person down with more force than that being exerted by the spring. If a person and a scale are in free fall, the scale reading would be zero because they are falling in unison and the scale is not pushing upwards on the person any longer.
 * Physics Talk: Forces Acting During Acceleration**

At rest/constant speed upward

1. The net force is 0. 2. The reading on the scale would be more than the person's actual weight. 3. When you accelerate upward, the floor or ground is pushing you upward with a greater force than that exerted by your weight. 4. The reading would be 0 because you and the scale would be falling in unison. 5. Air resistance slows falling raindrops.
 * Checking Up**

1a. 19.6 m/s every second (9.8 x 2) 1b. 49 m/s every second (9.8 x 5) 1c. 98 m/s every second (9.8 x 10) 2a. 3.2 m/s every second (1.6 x 2) 2b. 8 m/s every second (1.6 x 5) 2c. 16 m/s every second (1.6 x 10) 4. 5. The elevator is increasing downward/decreasing upward 6. The reading on the scale will be larger than the person's actual weight (600N) 7a. The bathroom scale's reading will decrease 7b. 50 kg = 490 N, 50 x 1.5= 75, 490- 75= 415 The scale will read the person's weight as 415 N. 8a. 490 N (N=w) 8b. N>w, Fnet=ma --> N-w=ma --> N-mg=ma --> N-490=50(2) --> 590 N 8c. 490 N (N=w) 9. First picture: the elevator and the man are both at rest/moving at constant speed making acceleration 0. The forces acting on the elevator and the man are balanced. Therefore, the scale gives an accurate reading of the man's weight. Second picture: the elevator and the man are both accelerating downward. The forces acting on the elevator and the man are unbalanced, the bigger force is gravity (moving in same direction as acceleration). Because the the man, the elevator, and the scale are falling in unison, the scale claims the man's weight is 0. Third picture: the elevator and the man are both accelerating upward. The forces acting on the elevator are unbalanced, the bigger force is the normal force exerted by the bathroom scale (moving in same direction as acceleration. Because the man and the scale are accelerating upwards, his scale claims his weight to be 100 N more than what he actually weighs.
 * Physics To Go**
 * = **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 ||

Prompt: Does your weight change when you are riding on a roller coaster? If you were sitting on a bathroom scale, would the scale give different readings at different places on the roller coaster? Response: Your actual weight does not change when you are riding on a roller coaster. However, your perception of weight- how light or heavy you feel- can be changed while riding on a roller coaster. This can be explained by physics. When you are accelerating upwards, you feel heavier than normal because the forces "pushing" you upwards are larger than the force of your weight. When you are accelerating downwards, you feel lighter than normal because the forces acting on you are lighter than the force of your weight. When you are at rest or moving at a constant speed you should feel normal because the forces are balanced (sum of all forces/Fnet=0). This reasoning explains why if you were on a scale if would give different readings at different places on the roller coaster, even though your actual weight never changes.
 * What do you think now?**

Section Seven
Prompt: Why don't you fall out of the roller coaster cart when it goes upside down during a loop? Response: Centripetal force prevents you from falling out of a roller coaster cart when it goes upside down during a loop
 * What do you think?**

Normal force is the force acting perpendicular to the surface. Centripetal force is any force directed towards the center that causes an object to follow a circular path at a constant speed. Centripetal force is ALWAYS directed toward the center. Examples of centripetal force: automobile moving around a curve has the force of friction between the tires and the road as centripetal force, Earth moving around the sun has a force of gravity towards the sun as centripetal force, clothes in a dryer have the walls of the dryer keeping the clothes moving in a circle (the water flies out in straight lines through the holes) as centripetal force. Centripetal force is NOT an additional force- it is the name given to a force like friction, tension, gravity, or the normal force when that force causes an object to move in a circle. The centripetal force could be a combination of any of these forces. The formula Fc=mv^2/r where Fc is the centripetal force, m is the mass of the object, v is the speed, and r is the radius of the circle is used to calculate centripetal force. Because of Newton's second law, if there is a net force, an object must be accelerating (Fnet=ma). In the case of circular motion this force (Fnet) is directed toward the center of the circle- this is called centripetal acceleration. Centripetal acceleration is the acceleration directed toward the center of a circle experienced by an object traveling in a circular path at constant speed. Although the centripetal force is always toward the center, the direction is always changing since in the cirlce, the centripetal force may be toward the left or the right or up, but still point toward the center. In a vertical loop, the centripetal force can be either the gravitational force, normal force of the track on the coaster car or a combination of the two. When it is a combination of the two, you must add the forces as vectors. At the bottom the the loop, the normal force points towards the center of the circle (upward) while the gravitational force points downward. This means that the normal force is greater than the gravitational force at the bottom of the loop/circle. The normal force corresponds to your apparent weight. The black force vectors shown in the above picture show the net centripetal force required to keep the object moving in a circular path at the top and bottom of the circle. The length of the black vectors is different because the object has a larger speed at the bottom of the loop- the directions are different because the centripetal force must point toward the center of the circle.
 * Physics Talk: Centripetal Force and Acceleration**

1. Centripetal force/acceleration is required to make an object travel in a circle. 2. If you are traveling in a circle at a constant speed, you are not accelerating. 3. At the top of a roller coaster loop, gravitational force (weight) and normal force both act downward toward the center of the loop and provide the centripetal force. 4. Normal force is responsible for your apparent weight on a roller coaster. 5. Centripetal force acting on an object is equal to the (mass of the object multiplied by the velocity squared) divided by the radius of the circle.
 * Checking Up**

Centripetal force= (mass)(acceleration) Centripetal acceleration= (velocity squared)/radius ^ Equation for both= Fc = (mass)(velocity squared)/radius
 * Class Notes 4/8/11**

Centripetal force/acceleration and mass- direct relationship Centripetal force/acceleration and velocity- direct square relationship Centripetal force/acceleration and radius- indirect (inverse) relationship

Safety- acceleration should be less than 4g

Apparent weight in circle corresponds to normal force.

Tangential speed- often remains constant.

1a. If the loose end of the string is held to the ground, the car will travel in a circular path. 1b. If the string were to break, the car would go off in a tangent from the circle. 2a. Friction between the tires and the road provides centripetal force. 2b. If the car were to hit a section of ice, it might venture off the circular path. 3. The faster she spins the key chain (velocity), the stronger the centripetal force acting on the key chain. 4. If there are more people in the car, the mass is larger, which means that the centripetal force will increase. 5. Fnet= sum of all forces on object, m= mass of object, v= velocity of object, r= radius of circle/curve 6a. The speed did not change. 6b. The velocity did change because the car changed directions. 6c. No change in magnitude- still traveling at 20m/s. Change in direction from east to north. 7. Acceleration is 2m/s per second towards the center. 13a. Bottom of hill #1- uncertain 13b. Top of vertical loop- lighter 13c. Bottom of vertical loop- heavier 13d. Bottom of hill #2- uncertain 13e. Lift hill (going up at constant speed)- no change 14a. Bottom of hill #1- zero 14b. Top of vertical loop- down 14c. Bottom of vertical loop- up 14d. Bottom of hill #2- zero 14e. Lift hill (going up at constant speed)- zero 14f. Horizontal loop- sideways 14g. Back curve- sideways
 * Physics To Go**

The seat exerts the most force at the bottom of the swing. At this position, normal force > gravitational force, because the bigger force/net force must be in the same direction as acceleration. As you rise, the contact force decreases. At the top of the swing, the contact force between the rider and the string should be 0 (at rest). The motion of the swing resembles a curve or circle, meaning that the net force will always point towards the center.
 * Extra Credit: Investigating Further**

1a. Centripetal force (Fnet) and mass have a direct relationship. When the mass gets larger, Fnet will become larger. 1b. Centripetal force (Fnet) and velocity have a direct square relationship. When the mass gets larger, Fnet will become larger. 2. If the velocity doubles, the force needed must quadruple. 3. Centripetal force (Fnet) and the radius have an inverse relationship. When the radius gets larger, Fnet will become smaller. 4. The larger the radius for the curve, the __less__ the force required to keep the car moving along the curve. If the curve is tight (radius is very small) then a __larger__ force is required. 5. When the string was made shorter, it was more difficult to keep the stopper moving in a circle. When the string was made longer, it required less effort to keep the stopper moving in a circle. 6a. Centripetal acceleration=(velocity squared)/radius Ac= (12^2)/20 Ac=144/20 Ac=7.2 Centripetal acceleration is 7.2 m/s ^2 6b. Centripetal force=(mass)(acceleration) Fc=(300)(7.2) Fc=2200 N Centripetal force must be 2200 N
 * Physics Plus**

Prompt: Why don't you fall out of the roller coaster cart when it goes upside down during a loop? Response:
 * What do you think now?**

Section Eight
Prompt: Does it take more energy to pull a roller coaster up a steep incline? Why is it more difficult to walk up a steep incline than a gentle incline? Response: I think it does take more energy (force) to move up a steep incline, as opposed to a gentler incline.
 * What do you think?**

Work is found by multiplying force and displacement, and is measured in joules. Power is work done divided by time elapsed and is measured in watts. Power is the speed at which work is done and energy is transferred.
 * Physics Talk: Work**

1. The energy is converted to GPE. 2. The roller coaster's GPE when it is at the top of the first hill comes from the work done to pull it up the incline. 3. Using a ramp reduces the force needed to move the objects. 4. When a roller coaster has stopped, its KE is 0. 5. Power is measured in watts.
 * Checking Up**

Section Nine
Prompt: "The Snake" roller coaster stays at ground level throughout the ride. The passengers move left, then right, then left again. Which parts of The Snake will be the most thrilling? If the speed of The Snake always remains the same, why will it still be fun? Response: The parts of The Snake involving change in direction will be the most thrilling. Although the snake always has the same speed, it does not always have the same velocity, providing thrills to the riders.
 * What do you think?**

Some quantities, like force, always have direction. Some quantities, like your age, never have direction. There are some quantities, like how fast you are traveling, that can include direction. A quantity with both magnitude and direction is called a vector. A quantity with magnitude but no direction is called scalar. Scalars are easy to add, subtract, multiply, and divide. Distance and speed are both scalar quantities. Displacement is a vector. To add vectors, you must draw them and use vector addition. Vectors are represented by arrows. When two vectors are perpendicular to each other, vector addition is an application of the Pythagorean theorem. Energy is a scalar quantity and all energies are measured in joules. Whatever the energy at the beginning of a roller coaster is, that is the energy at all times as long as friction is not significant. If two points on a roller coaster have the same height, they must have the same GPE. If they have the same GPE, they also have the same KE.
 * Physics Talk: Adding Scalars and Adding Vectors**

Key points: 1. The total mechanical energy (GPE + KE) is the same at every point as long as friction is not significant or motors do not add energy. 2. The GPE depends only on the height from a reference position (GPE = mgh) since the mass and the gravitational force remain the same. 3. If two points on a roller coaster have the same height, the roller coaster is moving at the same speed at those two points.

Energy considerations are path independent- it does not matter what happens between the places of interest, the energy at one point compared to the energy at another point will remain the same.

1. To add vector quantities, the vectors must be drawn and the Pythagorean theorem may need to be used. Vectors are represented by arrows in a free body diagram. 2. Energy is a scalar. Force is a vector. 3. The total energy must remain the same. If two points have the same height, they have the same GPE and therefore the same KE. The GPE depends only on the height since the mass and gravitational force remain the same. 4. The energy of the roller coaster does not depend on the coaster's path. 5. To provide a change in the energy of a roller coaster, work is required.
 * Checking Up**

2. They possess the same gravitational potential energy at the top. Because they start at the same height, they will have the same KE (and therefore velocity) at the bottom. 3a. Scalar 3b. Vector 3c. Scalar 3d. Vector 3e. Vector 3f. Vector 3g. Scalar 3h. Scalar 3i. Vector 4a. Scalar 4b.Vector 4c. Scalar 4d. Vector 5. The roller coaster applies KE and work to move up the hill. At the top the energy is converted into GPE. The energy is converted back into KE on the roller coaster's descent. Theoretically, the energy should remain the same throughout the ride, but due to friction the energy will actually decrease in real-life situations.
 * Physics To Go**

Prompt: "The Snake" roller coaster stays at ground level throughout the ride. The passengers move left, then right, then left again. Which parts of The Snake will be the most thrilling? If the speed of The Snake always remains the same, why will it still be fun? Response: The parts of The Snake that will be the most thrilling are those that involve acceleration. Though the speed always remains the same, The Snake involves changes in direction, which will also produce acceleration. The Snake can still be an enjoyable ride even without huge changes in speed and height.
 * What do you think now?**

Section Ten
Prompt: Does the knowledge that people can get hurt or die on a roller coaster change the thrill of the ride? Would your answer change if you found out that one-half of all roller coaster rides ended in the death of passengers? Response: Some people might find riding a notoriously dangerous roller coaster more thrilling. It would not be a smart choice to ride a coaster on which one-half of all riders have died.
 * What do you think?**

For a roller coaster to be considered safe, its acceleration must be less than 4 g's (39.2m/s^2). At the top of a vertical loop, acceleration must be greater than 1 g but still less than 4 g's. Amount of g's is found by dividing acceleration 9.8m/s^2 which is 1 g. Decreasing the speed/increasing the radius of a roller coaster will decrease the acceleration**.** Centripetal acceleration is found using the equation velocity squared divided by radius. Centripetal acceleration and net force are always in same direction- towards center of circle.
 * Physics Talk: Roller Coaster Safety**

1. The maximum acceleration on a safe roller coaster is 4 g's or 39.2m/s^2. 2. Decreasing height (therefore decreasing GPE and eventually speed) and increasing a radius can decrease acceleration. 3. The bottom of a loop has the greatest acceleration. 4. The bottom of a loop has the greatest normal force.
 * Checking Up**

2a. Initial height is 20.4m. 200m=mgh 200/9.8=20.4m 2b. Acceleration is 33.4m/s squared. a=400/12 2c. This is a safe acceleration because it is under 4g's. (33.4/9.8=3.4) 2d. At any speed above 22m/s would be unsafe on the roller coaster. 2e. Any speed above 17m/s would be unsafe. 3a. The acceleration is 62.5m/s squared. a=625/10 3b.This is not a safe acceleration because it is above 4 g's. (62.5/9.8=6.4) 6a. The centripetal acceleration is 8m/s squared. a=144/18 6b. The centripetal force is 7,200 N. (900)(8)=7200 6c. The roller coaster is rounding a curve, creating a centripetal force. (normal force of the track) 7a. The centripetal acceleration is 26.7m/s squared. a=400/15 7b. The centripetal force is 24000 N. (26.7)(900)=24000 7c. Yes, the roller coaster is safe. The force of the tracks is larger than the centripetal force. 8a. The centripetal acceleration will not change due to a change in mass. 8b. The roller coaster will be moving at the same speed. 8c. The roller coaster may need stronger tracks to accommodate the change in mass.
 * Physics To Go**

Prompt: Does the knowledge that people can get hurt or die on a roller coaster change the thrill of the ride? Would your answer change if you found out that one-half of all roller coaster rides ended in the death of passengers? Response: A roller coaster known as being dangerous may provide extra thrills for some people. However, this mentality should be considered in moderation- if one half of all rides ended in the death of passengers, it'd be hard to imagine any people wanting to go for a ride. Creating a "risky" reputation for a roller coaster should not be accomplished by building a defunct ride.
 * What do you think now?**