Dragons Descent: 1. ~13 seconds 2. ~2.5 second 3. a. gravity down = normal force (seat) up b. gravity down, the bar is also pushing you down c. gravity down < normal force (seat ) up d. gravity down < normal force (seat ) up 4. 5. 6. 7.
v = 53m/13s = 4.1 m/s v = 39m/2.5s = 15.6 m/s x = .5*9.8*2.52 = 30.6 m Yes. Falling at g, you would only go 30.6 meters. Since you went farther in the same time, the downward acceleration must have exceeded g.
8. a. At the start of the original ascent. At the start of the upward motion on each bounce. b. At the top of each ascent. c. Just before reaching the bottom on each bounce. d. At the start of each descent. 9. ???
Grand Prix Racers: 1. The seatbelt stops your forward motion when you brake hard (or run into something). The backrest pushes you forward when you press the accelerator. 2. You feel as if you are being pushed to the outside of the turn, but you really aren’t. Your body is continuing in a straight line, but the car is turning. The wall of the car pushes you in, toward the center of the turn. 3. The wall of the car. 4. Friction of the road on the tires. Fnormal car
center
Ffriction
Fgravity 5. 6. (Static) friction must apply a force in toward the center of the circle. The faster you go, the greater this force must be, but there is a maximum amount that static friction can apply. 7. The turns on race tracks are banked, so the normal force provides some of the force in toward the center.
Bumper Cars: 1. In a collision, the change in momentum of the car will equal the average force applied to the car times the time that force is applied. (Ft = mΔv) In order to make the force as small as possible, you would like the time of the collision to be as long as possible. Soft bumpers extend the duration of the collision and therefore make the ride safer. 2. Bouncy collisions make the change in momentum greater. This means the force on passengers is greater. Perhaps this is desirable, perhaps not. However, if the collisions weren’t elastic, the two cars would remain together and create more traffic jams! 3. a. Line 1 V11 = 0.53 m/s (this is impossible, as car one would have to move through car 2) b. Line 2 V11 = -0.40 m/s (the car bounces backwards) c. Line 3 V11 = 0.73 m/s d. Line 4 V21 = 0.0143 m/s (this one’s impossible, too, as it gains energy) 14. When car A hits and applies a force on car B, car B applies an equal and opposite force on car A. The forces on the cars must be equal, regardless of who is hitting or being hit.
Astrosphere: 1. a. b. c. d. e.
counterclockwise clockwise about halfway between the inner and outer most points friction with the seat on the outer edge of the seat
2. a. 2.7 m b. 2π*2.7*0.30 = 5.1 m/s 3. a. 3.7 – 2.7 = 1.0 m b. 3.7 + 2.7 = 6.4 m 4.
a. 2π*1.0*0.12 = 0.75 m/s b. 2π*6.4*0.12 = 4.8 m/s
5. a. 5.1 – 4.8 = 0.30 m/s (0.67 mph) b. 5.1 + 0.75 = 5.85 m/s (13 mph)
6. 5.1 + 4.8 = 9.9 m/s
Tilt-A-Whirl: 1. Each car completes one backward turn when going over a hill. 2. Chaotic 3. The car stays in its outermost position. (There might be other possibilities for 4 – 6) 4. Position yourself so you are on the downhill side at the start. 5. Time your shifts so you continue to rotate in one direction. 6. Do nothing – if the ride is at a medium speed. Wild Mouse 1. d = √25.0² + 39.8² = 47.0 m 2. A = L·w = (39.8 m)·(17.5 m) = 696.5 m² ~ 697 m² 3. V = L·w·h = (39.8 m)·(17.5 m)·(25.0 m) = 17412.5 m³ ~ 17400 m³ 4. d = 47.0 m = 47.0 m · (3.28 ft / 1 m) = 154.2 ft ~ 154. ft A = 697 m² = 697 m² · (3.28 ft / 1 m) · (3.28 ft / 1 m) = 7498.6048 ft² ~ 7500 ft² V = 17400 m³ (3.28 ft / 1 m)(3.28 ft / 1 m)(3.28 ft / 1 m) = 614003.4 ft³ ~ 614000 ft³ 5. PE = m g h = (500 kg) (9.80 m/s²) (25.0 m) = 122500 J 6. At least that much energy must be imparted during the first climb. 7. From the motor which runs the lift. Electrical energy is converted to rotational mechanical energy, which is converted to linear motion to lift the cars. 8. At least that much energy must be imparted during the first climb, but friction must be overcome along the way. Technically there must be enough energy imparted to also have the car in motion once it does reach the top or it would come to a stop. 9. final KE = initial PE ½ m v² = m g h v=√2gh
v = √(2 · 9.80 m/s² · 25.0 m) = √490 = 22.1 m/s v = 22.1 m/s · (2.24 mph / 1 m/s) = 49.527 mph ~ 49.5 mph 10. P = (122500 J) / (10 s) = 12250 W 11. I = P / V = (12250 W) / ( 220 V) = 55.682 A ~ 55.7 A 12. E = P · t = (12250 W) · (8 h) = 98000 W·h = 98 kWh 13. Cost = E × rate = (98 kWh) · ($0.18/kWh) = $17.64 14. Results vary. The value of t determined changes answers to 10 – 13. 15. Results vary. 16. Friction; Results vary; The difference between final kinetic energy and initial potential energy; Primarily through the non-conservative forces, primarily friction. Overcoming friction force f over a distance d requires energy equal to f·d. Casino: 1. Casino ride pattern
2. There are two axes of rotation. 3. You feel like you are lifted out of your seat slightly. Gravity is pulling down on you and the seat is pushing up. At the top, you must accelerate downwards, so the seat doesn’t push up on you as much as gravity pulls down. This makes you feel momentarily lighter. Flying Trapeze: 1. a. approximately 12 seconds b. 6 seconds
c. 0.17 rot/sec 2. a. 3.7sin(50) + 7.6 = 10.4 m b. 2 π*10.4 = 65.3 m c. 65.3/6 = 11.1 m/s 3. Ftension
Fgravity 4. Toward the center of the circle. 5. a. approximately 60 kg b. 711 N 6. a. tan(50) = (mv2/r)/mg b. so v = 11.0 m/s (very close to the measured value) 7. The top of the ride tilts, so the swings have to go up and down a little.
Sea Dragon: 1. a and b.
All the seats move at the same speed.
2. a. 6820 + 40*60 = 9220 kg b. 9220*9.8*7.0 = 632,000 J 3. a. (632,000*2/9220).5 = 11.7 m/s b. When the middle of the boat is at the bottom. 4. a. 6.28*(12.2/9.8).5 = 7.0 s b. 6 - 7 seconds c. well, but not perfectly 5. In the middle, horizontally. Vertically, it’s approximately 1 meter above the bottom.
6. The fact that the mass is spread out vertically would make the moment of inertia less than it would be for a point mass at a radius of 12.2 meters. A better prediction would be for the boat to move faster at the bottom. Additionally, the period would be less. 7. Once you measured the acceleration, you can use a = v2/r to find the speed. 8. The acceleration is less than g. You are not in freefall. If the center of the boat went to the height of the pivot, then the passengers would be in freefall.
Tempest in the Tea Cups: 1. a. Counterclockwise b. Clockwise 2. a. 6.28*2.1*0.30 = 4.0 m/s b. 6.28*5.8*0.16 = 5.8 m/s c. 4.0 = 1.8 m/s 3. a. b. c. approx. 2 seconds d. 6.28*.91 = 5.7 m e. 5.7/2 = 2.8 m/s 4. a. 2.8 + 1.8 = 4.6 m/s b. counterclockwise 5. The back of the seat. 6. Counterclockwise (in the same direction as the main platform)
Thunder Falls Log Flume: 1. B. Gravitational potential energy has increased significantly (and is at its maximum) C. Gravitational potential energy has decreased and kinetic energy has increased
D. Gravitational potential energy has decreased and kinetic energy has increased (and is at its maximum) E. Kinetic energy is still present, but has decreased. The water has gained kinetic energy and thermal energy. 2. The energy to raise the boat to B comes from the motor that runs the conveyor belt. The motor gets its energy from electricity. The power station that generates the electricity gets the energy (presumably) from fossil fuels. So, originally the energy was chemical potential energy in the fossil fuel. (There are some other alternatives.)
Excalibur: 1. a. (2*9.8*25.3).5 = 22.3 m/s = 49.7 mph 2. Thermal energy produced by friction. Also, the entire car does not drop 25.3 meters simultaneously. When one car is at the top, the others aren’t. When one car is at the bottom, the others aren’t. 3. Distance ID along track
h (m)
v (m/s)
A
0
B
53.4
1.3
21.9
C
122.3
19.9
9.4
D
208.3
10.5
15.7
E
307.3
12.9
12.5
F
406.6
0.0
19.2
G
504.8
2.2
17.0
H
539.0
12.6
8.5
I
609.5
3.7
14.8
J
676.7
6.9
11.6
PE (J)
26.6 -------- 1190000
1. a. 307,000 J
58100 889000 469000 576000 0 98300 563000 165000 308000
KE (J) --------
ETotal (J)
ELoss (J)
Tlen (m)
Avr. Loss (J/m)
1190000 -------- -------- --------
1090000 1150000
40000
53.4
749
201000
1090000
60000
68.9
871
562000
1030000
60000
86.0
698
356000
932000
98000
99.0
990
840000
840000
92000
99.3
926
659000
757000
83000
98.2
845
165000
728000
29000
34.2
848
499000
664000
64000
70.5
908
307000
615000
49000
67.2
729
b. 307,000/(.42*8*14*3*122*6) = 3.0˚C c. The brake pads also heat up. Since that takes some of the energy, The fins should heat up less.
