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A 50 kg box slides across a floor and its velocity decreases from 20 m/s to 5 m/s. a) What change in Kinetic Energy (ฮKE) did the box undergo to reduce its velocity? ฮKE = ยฝ (m)(vf
2 โ vi2)
ฮKE = ยฝ (50 kg)[(5 m/s)2 โ (20 m/s)2]
ฮKE = ยฝ (50 kg)[-375 m2/s2]
ฮKE = -9375 kgm2/s2
ฮKE = -9375 J
A 50 kg box slides across a floor and its velocity decreases from 20 m/s to 5 m/s. b) How much work does the frictional resistance of the floor do upon the block?
Work Done by Frictional Forces Is EQUAL and OPPOSITE
The Change in Kinetic Energy
+9375 J
c) If the reduction in speed takes place over 20 m, what is the force of friction?
Work (J) = Force (N) X distance (m)
Force (N)= [Work (J) / distance
(m)]
NNForce
m
JNForce
mcedis
JWorkNForce
469)(
20
9375)(
)(tan
)()(
Exercise 1:On his way off to college, Russell drags his suitcase 15.0 m from the door of his house to the car at a constant speed with a horizontal force of 95.0 N.a) How much work does Russell do to
overcome the force of friction?b) If the floor has just been waxed, does he have to do more work or less work to move the suitcase? Explain.
๐ยฟ๐พ=๐ญ โ๐ =(๐๐ .๐๐ต ) (๐๐ .๐๐ )=๐๐๐๐ ๐ฑ
Exercise 2: Katie, a 30.0-kg child, climbs a tree to rescue her cat who is afraid to jump 8.0 m to the ground. How much work does Katie do in order to reach the cat?
Exercise 3: Marissa does 3.2 J of work to lower the window shade in her bedroom adistance of 0.8 m. How much force must Marissa exert on the window shade?
Exercise 4: Atlas and Hercules, two carnival sideshow strong men, each lift 200. kg barbells 2.00 m off the ground. Atlas lifts his barbells in 1.00 s and Hercules lifts his in 3.00 s.
a) Which strong man does more work? man is more powerful.
Hercules and AtlasโฆSame Workโฆ
Exercise 4: Atlas and Hercules, two carnival sideshow strong men, each lift 200. kg barbells 2.00 m off the ground. Atlas lifts his barbells in 1.00 s and Hercules lifts his in 3.00 s.
b) Calculate which man is more powerful.
Hercules
Atlas
Example 7: Frank, a San Francisco hot dog vender, has fallen asleep on the job. When an earthquake strikes, his 300 kg hot dog cart rolls down Nob Hill and reaches point A at a speed of 8.00 m/s. How fast is the hot-dog cart going at point B when Frank finally wakes up and starts to run after it?
๐ฒ๐ฌ๐+๐ท๐ฌ๐=๐ฒ๐ฌ๐ +๐ท๐ฌ๐
(๐๐ )๐๐๐๐+๐๐๐๐=(๐๐ )๐๐ ๐ ๐+๐๐๐ ๐
Mass is same throughout and can be factored outโฆ
(๐๐ )๐ ๐๐+๐๐๐=(๐๐ )๐ ๐ ๐+๐๐ ๐
Solve for vf
=
=
Exercise 5:It is said that Galileo dropped objects off the Leaning Tower of Pisa to determine whether heavy or light objects fall faster. If Galileo had dropped a 5.0 kg cannon ball to the ground from a height of 12 m, what would have been the change in PE of the cannon ball?
โ ๐ท๐ฌ=๐๐๐โ ๐ท๐ฌ=(๐๐๐)(๐๐๐
๐๐ )(๐๐๐)
โ ๐ท๐ฌ=๐๐๐ ๐ฑ
Exercise 6: On June 5, 2007, Rags to Riches, became the first filly to win the Belmont Stakes horse race since 1905, running with an average speed of 16.23 m/s. If Rags to Riches and jockey Johnny Velazquez had a combined mass of 550.0 kg, what was their KE as they crossed the finish line?
โ ๐ฒ๐ฌ=๐๐
๐๐๐
โ ๐ฒ๐ฌ=๐๐
(๐๐๐๐๐)(๐๐ .๐๐๐๐ )
๐
โ ๐ฒ๐ฌ=๐๐ ,๐๐๐ ๐ฑ
Exercise 7: Brittany is changing the tire of her car on a steep hill 20.0 m high. She trips and drops the 10.0-kg spare tire, which rolls down the hill with an initial speed of 2.00 m/s. What is the speed of the tire at the top of the next hill, which is 5.00 m high?
(๐๐ )๐๐๐๐+๐๐๐๐=(๐๐ )๐๐ ๐ ๐+๐๐๐ ๐
=
=
=
Exercise 8: A Mexican jumping bean jumps with the aid of a small worm that lives inside the bean. a) If a bean of mass 2.0 g jumps 1.0 cm from your hand into the air, how much potential energy has it gained in reaching its highest point. b) What is its speed as the bean lands back in the palm of your hand?
mv2
=
โ ๐ท๐ฌ=(๐ .๐๐๐๐ค๐ )(๐๐ ๐๐๐ )(๐ .๐๐๐ฆ)
J =
๐=โยฟ ยฟ
Exercise 9: A 500.-kg pig is standing at the top of a muddy hill on a rainy day. The hill ha a vertical drop of 30.0 m. The pig slips and begins to slide down the hill. What is the pigโs speed at the bottom of the hill? Use the law of conservation of energy.
๐=๐๐ .๐๐/๐
โ ๐ฒ๐ฌ=โ๐ท๐ฌ
๐=โ๐๐๐ ๐=โ๐ (๐๐๐๐๐ )(๐๐๐)
๐๐
๐๐=๐๐๐๐
๐๐๐=๐๐๐
Exercise 10: While on the moon, the Apollo astronauts enjoyed the effects of a gravity much smaller than that on Earth. If Neil Armstrong jumped up on the moon with an initial speed of 1.51 m/s to a height of 0.700 m, what amount of gravitational acceleration did he experience?
๐=(๐ .๐๐๐ /๐ )๐
๐ (๐ .๐๐๐๐)
โ ๐ฒ๐ฌ=โ๐ท๐ฌ๐๐
๐๐=๐๐๐๐
๐๐๐=๐๐๐ ๐๐
๐๐=๐
The Sweet SpotThe sweet spot of a softball bat or a tennis racquet is the place where the ballโs impact produces minimum vibrations in the racquet or bat. Strike a ball at the sweet spot and it goes faster and farther. Strike a ball in another part of the bat or racquet, and vibrations can occur that sting your hand! From an energy point of view, there is energy in the vibrations of the bat or racquet. There is energy in the ball after being struck. Energy that is not in vibrations is energy available to the ball. Do you see why a ball will go faster and farther when struck at the sweet spot?
Due to friction, energy is transferred both into the floor and into the tire when the bicycle skids to a stop.
a. An infrared camera reveals the heated tire track on the floor.
b. The warmth of the tire is also revealed.
Typical stopping distances for cars equipped with antilock brakes traveling at various speeds. The work done to stop the car is friction force distance of slide. (Miles per Hour)
28
56
113
(Slide is in meters)10
401 6 0
What process provides energy for rockets that lift the space shuttle into orbit? What process releases energy from the food we eat? The answer is chemical reactions.
During a chemical reaction the bonds between atoms break and then reform. Breaking bonds requires energy, and forming bonds releases it. Pulling atoms apart is like pulling apart two magnets stuck together; it takes energy to do it. And when atoms join, it is like two separated magnets that slam together; energy is released.
Rapid energy re lease can produce flames. Slow energy re lease occurs during the digestion of food. The conservation of energy rules chemical reactions. The amount of energy required to break a chemical bond is the same amount released when that bond is formed.
The potential energy of the 100-N boulder with respect to the ground below is 200 J in each case because the work done in elevating it 2 m is the same whether the boulder isโฆa. Lifted with
100 N of forceโฆ
No work is done in moving it horizontally, neglecting friction.
Potential Energy
KE (J) PE (J) ฮฃFw
(N)d
(m)
500200
010,00
010,00
0
500150
2,500 7,50010,00
0
500100
5,000 5,00010,00
0
500 50 7,500 2,50010,00
0
500 0 10,000 010,00
0
Everywhere along the path of the pendulum bob, the sum of
PE and KE is the same. Because of the work done
against friction, this energy will eventually be transformed
into heat.
b. Pushed up the 4-m incline with 50 N of force.
No work is done in moving it horizontally, neglecting friction.
C. Lifted with 100 N of force up each 0.5-m stair.
No work is done in moving it horizontally, neglecting friction.
Energy Conservation Most energy consumed in America comes from fossil fuels. Oil, natural gas, and coal supply the energy for almost all our industry and technology. About 70% of electrical power in the United States comes from fossil fuels, with about 21% from nuclear power. Worldwide, fossil fuels also account for most energy consumption. We have grown to depend on fossil fuels because they have been plentiful and inexpensive. Until recently, our consumption was small enough that we could ignore their environmental impact.
But things have changed. Fossil fuels are being consumed at a rate that threatens to deplete the entire world supply. Locally and globally, our fossil fuel consumption is measurably polluting the air we breathe and the water we drink. Yet, despite these problems, many people consider fossil fuels to be as inexhaustible as the sunโs glow and as acceptable as Momโs apple pie, because these fuels lasted and nurtured us through the 1900s.
Financially, fossil fuels are still a bargain, but this is destined to change. Environmentally, the costs are already dramatic. Some other fuel must take the place of fossil fuels if we are to maintain the industry and technology to which we are accustomed. The French have chosen nuclear, with about 74% of their electricity coming from nuclear power plants. What energy source would you choose as an alternative?
In the meantime, we shouldnโt waste energy. As individuals, we should limit the consumption of useful energy by such measures as turning off unused electrical appliances, using less hot water, going easy on heating and air conditioning, and driving energy-efficient automobiles. By doing these things, we are conserving useful energy.
When electric current passes through water, bubbles of hydrogen form at one wire and bubbles of oxygen form at the other.
In a fuel cell, the reverse process occurs: hydrogen and oxygen combine to produce water and electricity.
Exercise 11: Cathy, a 460-N actress playing Peter Pan, is hoisted above the stage in order to โflyโ by a stagehand pulling with a force of 60. N on a rope wrapped around a pulley system. What is the actual mechanical advantage of the pulley system?
๐จ๐ด๐จ=๐ญ ๐
๐ญ ๐
๐จ๐ด๐จ=๐๐๐๐ต๐๐๐ต
๐จ๐ด๐จ=๐ .๐
Exercise 12: A windmill uses sails blown by the wind to turn an axle that allows a grindstone to grind corn into meal with a force of 90. N. The windmill has sails of radius 6.0 m blown by a wind that exerts a force of 15 N on the sails, and the axle of the grindstone has a radius of 0.50 m. a) What is the ideal mechanical advantage of the wheel?
๐ฐ๐ด๐จ=๐ ๐
๐ ๐
๐ฐ๐ด๐จ=๐๐ ๐ ๐
๐๐ ๐๐
๐ฐ๐ด๐จ=๐ ๐
๐ ๐๐ฐ๐ด๐จ=
๐ .๐๐.๐๐
๐ฐ๐ด๐จ=๐๐
Exercise 12: A windmill uses sails blown by the wind to turn an axle that allows a grindstone to grind corn into meal with a force of 90. N. The windmill has sails of radius 6.0 m blown by a wind that exerts a force of 15 N on the sails, and the axle of the grindstone has a radius of 0.50 m. b) What is the actual mechanical advantage of the wheel?
๐จ๐ด๐จ=๐ญ ๐
๐ญ ๐๐จ๐ด๐จ=
๐๐๐ต๐๐๐ต
๐จ๐ด๐จ=๐
Exercise 12: A windmill uses sails blown by the wind to turn an axle that allows a grindstone to grind corn into meal with a force of 90. N. The windmill has sails of radius 6.0 m blown by a wind that exerts a force of 15 N on the sails, and the axle of the grindstone has a radius of 0.50 m. c) What is the efficiency of the wheel?
๐ฌ ๐๐๐ข๐๐ข๐๐ง๐๐ฒ=๐จ๐ด๐จ๐ฐ๐ด๐จ
๐ฌ ๐๐๐ข๐๐ข๐๐ง๐๐ฒ=๐๐๐
๐ฌ ๐๐๐ข๐๐ข๐๐ง๐๐ฒ=๐ .๐
Exercise 13: Winnie, a waitress, holds in one hand a 5.0-N tray stacked with twelve 3.5-N dishes. The length of her arm from her hand to her elbow is 30.0 cm and her biceps muscle exerts a force 5.0 cm from her elbow, which acts as a fulcrum. How much force must her biceps exert to allow her to hold the tray?Assuming an efficiency of
1โฆ
=๐ ๐
๐ ๐=
๐ญ๐
๐ญ ๐
=
=
Exercise 14: When building the pyramids, the ancient Egyptians were able to raise large stones to very great heights by using inclines. If an incline has an ideal mechanical advantage of 4.00 and the pyramid is 15.0 m tall, how much of an angle would the incline need in order for the Egyptian builder to reach the top? ๐ ๐=๐ ๐ ๐ฐ๐ด๐จ
๐ ๐=(๐๐๐)(๐)๐ ๐=๐๐๐
๐ ๐=๐๐๐
๐ ๐=๐๐๐
๐ฝ=๐๐๐โ๐ ๐๐๐๐๐๐
๐ฝ=๐๐ .๐๐
๐
Exercise 15: The Ramseys are moving to a new town, so they have called in the ACME moving company to take care of their furniture. Debbie, one of the movers, slides the Ramseysโ 2200-N china cabinet up a 6.0-m-long ramp to the moving van, which stands 1.0 m off the ground.
a) What is the ideal mechanical advantage of the incline?
IMA
๐ ๐=๐๐
๐
IMA
IMA๐
๐=๐๐
b) If Debbie must exert a 500.-N force to move the china cabinet up the ramp with a constant speed, what is the actual mechanical advantage of the ramp?
A-1: On a ski weekend in Colorado, Bob, whose mass is 75.0 kg, skis down a hill that is inclined at an angle of 15.0ยฐ to the horizontal and has a vertical rise of 25.0 m. How much work is done by gravity on Bob as he goes down the hill?
๐=๐ญ๐
๐=๐๐๐๐๐ ๐ฑ
๐=๐๐๐ ๐=๐๐๐๐ (๐๐๐
๐๐ )๐๐ .๐๐
25.0 m.
c) What is the efficiency of the ramp?
๐ฌ๐๐ =๐จ๐ด๐จ๐ฐ๐ด๐จ
๐ฌ๐๐ =๐ .๐๐
๐ฌ๐๐ =.๐๐
A-2: A pile driver is a device used to drive stakes into the ground. While building a fence, Adam drops a pile driver of mass 3000. kg through a vertical distance of 8.0 m. The pile driver is opposed by a resisting force of 5.0 x 106 N. How far is the stake driven into the ground on the first stroke?
๐ญ ๐๐๐๐ ๐๐๐=๐๐๐
๐ ๐๐๐= .๐๐๐๐
๐ ๐๐๐=๐๐๐๐๐๐(๐๐๐
๐๐ )๐๐
๐๐๐๐๐๐ต
A-3: At an amusement park, a ride called the โCycloneโ is a giant roller coaster that ascends a 34.1-m hill and then drops 21.9 m before ascending the next hill. The train of cars has a mass of 4727 kg.
a) How much work is required to get an empty train of cars from the ground to the top of the first hill?
๐=๐๐๐
J
(34.1 m)
b) What power must be generated to bring the train to the top of the first hill in 30.0 s?
๐ท ,๐พ๐๐๐ ๐ฌ=๐๐๐๐ , ๐ฑ๐๐๐๐ ,๐
Watts
๐ท=๐ .๐ ๐๐๐๐ J ๐๐ ๐
c) How much PE is converted into KE from the top of the first hill to the bottom of the 21.9 m drop?
)
(21.9 m)
J
A-4 A flea gains 1.0 x 10-7 J of PE jumping up to a height of 0.030 m from a dogโs back. What is the mass of the flea?
โ ๐ท๐ฌ=๐ฆ๐ ๐
kg
๐=โ๐ท๐ฌ๐๐
๐= ๐ ๐๐๐โ๐ ๐ฑ
(๐๐๐๐๐ )( .๐๐๐๐ยฟ
ยฟ
A-5: At target practice, Diana holds her bow and pulls the arrow back a distance of 0.30 m by exerting an average force of 40.0 N. What is the potential energy stored in the bow the moment before the arrow is released?
๐ท๐ฌ=๐ ๐๐ท๐ฌ=(๐๐๐ )(๐ .๐๐ฆ )
๐ท๐ฌ=๐๐ ๐
A-6: The coyote, whose mass is 20.0 kg, is chasing the roadrunner when the coyote accidentally runs off the edge of a cliff and plummets to the ground 30.0 m below. What force does the ground exert on the coyote as he makes a coyote-shaped dent 0.420 m deep in the ground?
๐๐๐=๐ ๐
๐ญ=๐๐ ,๐๐๐ ๐
(๐๐๐๐ )(๐๐๐๐๐ )(30 m )=๐ (๐ .๐๐๐๐ฆ)
A-7: A 0.080-kg robin, perched on a power line 6.0 m above the ground, swoops down to snatch a worm from the ground and then returns to an 8.0-m-high tree branch with his catch. a) By how much did the birdโs PE increase in its trip from the power line to the tree branch?
โ ๐ท๐ฌ=๐๐โ๐
โ ๐ท๐ฌ=(๐ .๐๐๐๐ )(๐๐ ๐๐๐ )( 2 m )
โ ๐ท๐ฌ=๐.๐ ๐ฑ
b) How would your answer have changed if the bird had flown around a bit before landing on the tree branch?
A-8: Cinnamon, whose mass is 5.45 kg, was adopted from a cat shelter and now enjoys napping on top of the refrigerator. Cinnamon rolls over and falls off the refrigerator landing feet first with a KE of 85.5 J as she hits. How tall is the refrigerator?
โ ๐ท๐ฌ=โ๐ฒ ๐ฌ
โ๐=๐ .๐๐๐
๐๐โ๐=โ๐ฒ ๐ฌ
(๐ .๐๐๐๐ )(๐๐๐๐๐ )โ๐=๐๐ .๐ ๐ฑ
A-9: Calories measure energy we get from food, and one dietary Calorie is equal to 4187 J. The average food energy intake for human beings is 2000. Calories/day. Assume you have a mass of 55.0 kg and you want to burn off all the Calories you consume in one day. How high a mountain would you have to climb to do so?
โ ๐ท๐ฌ=โ๐ฒ ๐ฌ4187 J)(One Day)
๐บ๐๐๐๐ ๐๐๐ โ๐โ๐=
(๐๐๐๐๐ช๐๐ )(4187 J)
(๐ .๐๐๐๐ )(๐๐๐๐๐ ) 15,200 m
๐๐โ๐=โ๐ฒ ๐ฌ
A-10: From a height of 2.15 m above the floor of the basketball court, forward Paul Pierce tosses a shot straight up next to the basketball hoop with a KE of 5.40 J. If his regulation-size basketball has a mass of 0.600 kg, will his shot go as high as the 3.04-m hoop? Use the law of conservation of energy.
โ ๐ท๐ฌ=โ๐ฒ ๐ฌ J)๐บ๐๐๐๐ ๐๐๐ ๐ ๐
+ 2.15 m3.07 m
๐๐โ๐=โ๐ฒ ๐ฌ
๐ ๐ญโฒ ๐ฌ๐ ๐จ๐ญ ๐ญ๐ก๐ ๐ฃ๐ฎ๐ข๐๐ !
A-11: Mr. Macintosh, a computer technician, uses a screwdriver with a handle of radius 1.2 cm to remove a screw in the back of a computer. The screw moves out 0.20 cm on each complete turn. What is the ideal mechanical advantage of the screwdriver?
๐ฐ๐ด๐จ=๐ ๐
๐ ๐
๐ฐ๐ด๐จ=๐ .๐๐๐๐ .๐๐๐๐
๐ฐ๐ด๐จ=๐
A-12: Tomโs favorite pastime is fishing. a) How much work is required for Tom to reel in a 10.0-kg bluefish from the waterโs surface to the deck of a fishing boat, 5.20 m above the water, if the reel of his fishing pole is 85.0% efficient?
Eff .
๐ ๐=๐๐๐๐ (๐๐๐
๐๐ )๐ .๐๐
๐ .๐๐๐ ๐=๐๐๐ ๐ฑ
b) If Tom applies a force of 15 N to the reelโs crank handle, what is the actual mechanical advantage of the fishing pole?
๐จ๐ด๐จ=๐ญ ๐
๐ญ ๐๐จ๐ด๐จ=
๐๐๐๐ต๐๐๐ต
๐จ๐ด๐จ=๐ .๐
c) What is the ideal mechanical advantage of the fishing pole?
๐ฌ๐๐ =๐จ๐ด๐จ๐ฐ๐ด๐จ
๐ฐ๐ด๐จ=๐จ๐ด๐จ๐ฌ๐ญ๐ญ
๐ฐ๐ด๐จ=๐ .๐.๐๐
๐ฐ๐ด๐จ=๐ .๐
A -13 A nutcracker 16 cm long is used to crack open a Brazil nut that is placed 12 cm from where your hand is squeezing the nutcracker. What is the ideal mechanical advantage of the nutcracker?
๐ฐ๐ด๐จ=๐ ๐
๐ ๐
๐ฐ๐ด๐จ=๐
๐ฐ๐ด๐จ=๐๐๐๐๐๐๐
B-1: A 5.00-N salmon swims 20.0 m upstream against a current that provides a resistance of 1.50 N. This portion of the stream rises at an angle of 10.0ยฐ with respect to the horizontal. a) How much work is done by the salmon against the current?
๐ ๐ =๐ญ โ๐
B-1: A 5.00-N salmon swims 20.0 m upstream against a current that provides a resistance of 1.50 N. This portion of the stream rises at an angle of 10.0ยฐ with respect to the horizontal. b) What is the gain in PE by the salmon?
sinฮธ)
do
di = 20 m
ฮธ = 100
)
โ ๐ท๐ฌ=๐๐๐ ๐ฑ
B-1: A 5.00-N salmon swims 20.0 m upstream against a current that provides a resistance of 1.50 N. This portion of the stream rises at an angle of 10.0ยฐ with respect to the horizontal. c) What is the total work that must be done by the salmon?
๐ ๐ป๐๐๐๐=๐๐ .๐ ๐ฑ+๐๐ ๐ฑ
๐ ๐ป๐๐๐๐=๐๐ .๐ ๐ฑ
๐ ๐ป๐๐๐๐=โ๐ท๐ฌ+๐ ๐
B-1: A 5.00-N salmon swims 20.0 m upstream against a current that provides a resistance of 1.50 N. This portion of the stream rises at an angle of 10.0ยฐ with respect to the horizontal. d) If the salmon takes 40.0 s to swim the distance, what power does it exert in doing so?
๐ท๐๐๐๐ ,๐พ=๐พ๐๐๐ , ๐ฑ๐ป๐๐๐ ,๐
๐ท๐๐๐๐ ,๐พ=๐๐ .๐ ๐ฑ๐๐ ๐ P
B-2: A 30-kg shopping cart full of groceries sitting at the top of a 2.0-m hill begins to roll until it hits a stump at the bottom of the hill. Upon impact, a 0.25-kg can of peaches flies horizontally out of the shopping cart and hits a parked car with an average force of 490 N. How deep a dent is made in the car?
๐ญ๐ =๐๐๐ ๐ =๐๐๐
๐ญ
๐ =.๐๐๐๐(๐๐๐
๐๐ )๐๐
๐๐๐๐ต๐ =๐ .๐๐๐๐
B-3: Using her snowmobile, Midge pulls a 60.0 kg skier up a ski slope inclined at an angle of 12.0ยฐ to the horizontal. The snowmobile exerts a force of 200. N parallel to the hill. If the coefficient of friction between the skis and the snow is 0.120, how fast is the skier moving after he has been pulled for 100.0 m starting from rest?
ฮธ = 120
60 kg
Fi = 200 N
ฮผ = 0.12d
i = 200 N do
B-4: Jose, whose mass is 45.0 kg, is riding his 5.0-kg skateboard down the sidewalk with a constant speed of 6.0 m/s when he rolls across a 10.0-m-long patch of sand on the pavement. The sand provides a force of friction of 6.0 N. What is Joseโs speed as he emerges from the sandy section?
๐ฏ=โ(๐๐ )๐โ(๐๐ญ๐ ๐ )
v v
B-5: Eben lifts an engine out of his car with the help of a winch that allows him to raise the engine 0.020 m for every 0.90 m he pulls on the cable. Eben expends 1000. J of energy to lift the 800.-N engine 0.50 m. a) What is the efficiency of the winch?
๐ฌ๐๐ =๐ญ๐ ๐ ๐
๐ ๐
๐ฌ๐๐ =(๐๐๐๐ต )๐ .๐๐๐ยฟ ยฟ๐๐๐๐ ๐ฑ
๐ฌ๐๐ =๐ .๐
B-5: Eben lifts an engine out of his car with the help of a winch that allows him to raise the engine 0.020 m for every 0.90 m he pulls on the cable. Eben expends 1000. J of energy to lift the 800.-N engine 0.50 m. b) What is the ideal mechanical advantage of the winch?
๐๐๐=๐ ๐
๐ ๐๐๐๐=
๐ .๐๐๐ .๐๐๐
๐๐๐=๐๐
B-5: Eben lifts an engine out of his car with the help of a winch that allows him to raise the engine 0.020 m for every 0.90 m he pulls on the cable. Eben expends 1000. J of energy to lift the 800.-N engine 0.50 m. c) What is the actual mechanical advantage of the winch?
๐๐๐=(๐ฐ๐ด๐จ)(๐ฌ๐๐ )๐๐๐=(๐๐)(๐ .๐)
๐๐๐=๐๐
B-5: Eben lifts an engine out of his car with the help of a winch that allows him to raise the engine 0.020 m for every 0.90 m he pulls on the cable. Eben expends 1000. J of energy to lift the 800.-N engine 0.50 m. d) What force does Eben exert to lift the engine?
๐ญ ๐๐=๐ญ๐๐๐
๐จ๐ด๐จ๐ญ ๐๐=
๐๐๐๐ต๐๐๐ญ ๐๐=๐๐๐ต
1. A force sets an object in motion. When the force is multiplied by the time of its application, we call the quantity impulse, which changes the momentum of that object. What do we call the quantity force distance, and what quantity can this change?2. Work is required to lift a barbell. How many times more work is required to lift the barbell three times as high?3. Which requires more work, lifting a 10-kg load a vertical distance of 2 m or lifting a 5-kg load a vertical distance of 4 m?1. Work; Objectโs Energy2. Three3. Both the same, 200 J
4. How many joules of work are done on an object when a force of 10 N pushes it a distance of 10 m?5. How much power is required to do 100 J of work on an object in a time of 0.5 s? How much power is required if the same work is done in 1 s?6. What are the two main forms of mechanical energy?4. 5.
6. PE and KE
7. a. If you do 100 J of work to elevate a bucket of water, what is its gravitational potential energy relative to its starting position?b. What would the gravitational potential energy be if the bucket were raised twice as high?8. A boulder is raised above the ground so that its potential energy relative to the ground is 200 J. Then it is dropped. What is its kinetic energy just before it hits the ground?7. a. 100 J b. 200 J8. 200 J
9. Suppose you know the amount of work the brakes of a car must do to stop a car at a given speed. How much work must they do to stop a car that is moving four times as fast? How will the stopping distances compare?10. How does speed affect the friction between a road and a skidding tire?11. What will be the kinetic energy of an arrow having a potential energy of 50 J after it is shot from a bow? 9. 16 times as much work 16 times the distance10. Speed does not affect friction.11. 50 J
12. What does it mean to say that in any system the total energy score stays the same?13. In what sense is energy from coal actually solar energy?14. How does the amount of work done on an automobile by its engine relate to the energy content of the gasoline?15. In what two ways can a machine alter an input force?12. Energy is conserved.13. Material that forms coal was produced by sunโs energy.14. It is less than the energy in the gasoline.15. It can change its magnitude and/or direction
16. In what way is a machine subject to the law of energy conservation? Is it possible for a machine to multiply energy or work input?17. What does it mean to say that a machine has a certain mechanical advantage? 18. In which type of lever is the output force smaller than the input force?19. What is the efficiency of a machine that requires 100 J of energy to do 35 J of work?16. Work out cannot exceed work in; no.17. It can multiply force by a certain amount.18. Type 3โalways, Type 1โ maybe19. 35%
20. Distinguish between theoretical mechanical advantage and actual mechanical advantage. How would these compare if a machine were 100% efficient?21. What is the efficiency of her body when a cyclist expends 1000 W of power to deliver mechanical energy to the bicycle at the rate of 100 W?22. In what sense are our bodies machines?20. TMAโno friction AMAโwith friction; same21. 10%22. Like machines, our bodies need an energy supply. Also, the same principle of combustion occurs in the metabolism of food in the body and the burning of fossil fuels.
23. What is the ultimate source of the energy derived from the burning of fossil fuels, from dams, and from windmills?24. What is the ultimate source of geothermal energy?25. Can we correctly say that a new source of energy is hydrogen? Why or why not?23. The sun24. Radioactivity in Earthโs interior25. No. Hydrogen is not a new source of energy because it takes energy to extract hydrogen from water and carbon compounds.
26. The mass and speed of three vehicles are shown below.a. Rank the vehicles by momentum from greatest to least.b. Rank the vehicles by kinetic energy from greatest to least.26. a. B, A, C b. C, B, A
27. Consider these four situations.(A) a 3-kg ball at rest atop a 5-m-tall hill(B) a 4-kg ball at rest atop a 5-m-tall hill(C) a 3-kg ball moving at 2 m/s atop a 5-m-tall hill(D) a 4-kg ball moving at 2 m/s at ground levela. Rank from greatest to least the potential energy of each ball.b. Rank from greatest to least the kinetic energy of each ball.c. Rank from greatest to least the total energy of each ball.27. a. B, A = C, D b. D, C, A = B c. B, C, A, D
28. A ball is released at the left end of the metal track shown below. Assume it has only enough friction to roll, but not to lessen its speed.a. Rank from greatest to least the ballโs momentum at each point.b. Rank from greatest to least the ballโs kinetic energy at each point.c. Rank from greatest to least the ballโs potential energy at each point.
28. a. C, B = D, A b. C, B = D, A c. A, B = D, C
29. The roller coaster ride starts with the car at rest at point A.a. Rank from greatest to least the carโs speed at each point.b. Rank from greatest to least the carโs kinetic energy at each point.c. Rank from greatest to least the carโs potential energy at each point.29. a. D, B, C, E, A b. D, B, C, E, A c. A, E, C, B, D
30. Rank the efficiency of these machines from highest to lowest.(A) energy in 100 J; energy out 60 J(B) energy in 100 J; energy out 50 J(C) energy in 200 J; energy out 80 J(D) energy in 200 J; energy out 120 J30. A = D, B, C
31. Carts moving along the lab floor run up short inclines. Friction effects are negligible.a. Rank the carts by kinetic energy before they meet the incline.b. Rank the carts by how high they go up the incline.c. Rank the carts by potential energy when they reach the highest point on the incline.d. Why are your answers different for b and c?31. a. B, A, C, D b. A, B, C, D c. B, A, C, D d. PETop = KEBottom
32. Rank the scale readings from greatest to least. (Ignore friction.) 33. Calculate the work done when a force of 1 N moves a book 2 m.34. Calculate the work done when a 20-N force pushes a cart 3.5 m.35. Calculate the work done in lifting a 500-N barbell 2.2 m above the floor. (What is the potential energy of the barbell when it is lifted to this height32. 33. 34. 35.
36. Calculate the watts of power expended when a force of 1 N moves a book 2 m in a time interval of 1 s.37. Calculate the power expended when a 20-N force pushes a cart 3.5 m in a time of 0.5 s.
38. How many joules of potential energy does a 1-kg book gain when it is elevated 4 m? When it is elevated 8 m? 36. 37. 38. = (1 kg)(10 N/kg)(4 m) = 40 J =80 J
39. Calculate the increase in potential energy when a 20-kg block of ice is lifted a vertical distance of 2 m.40. Calculate the number of joules of kinetic energy a 1-kg book has when tossed across the room at a speed of 2 m/s.41. How much work is required to increase the kinetic energy of a car by 5000 J?42. What change in kinetic energy does an airplane experience on takeoff if it is moved a distance of 500 m by a sustained net force of 5000 N?39. 40. 41. 42.
43. Which requires more work: stretching a strong spring a certain distance or stretching a weak spring the same distance? 44. Two people who weigh the same amount climb a flight of stairs. The first person climbs the stairs in 30 s, while the second person climbs them in 40 s. Which person does more work? Which uses more power?43. More force to stretch strong spring, so more work in stretching the same distance.44. Same work done by each, for same hour; climber in 30 s uses more power due to shorter time.
45. A physics teacher demonstrates energy conservation by releasing a heavy pendulum bob, as shown in the sketch, allowing it to swing to and fro. What would happen if, in his exuberance, he gave the bob a slight shove as it left his nose? Explain.46. Consider the kinetic energy of a fly in the cabin of a fast-moving train. Does it have the same or different kinetic energies relative to the train? Relative to the ground outside?45. If ball is given an initial KE, it returns to its starting position with that KE (moving in the other direction!) and hits the instructor.46. Just as motion is relative, KE is also. The speed and KE of the fly are different relative to the train and the ground.
47. When a driver applies brakes to keep a car going downhill at constant speed and constant kinetic energy, the potential energy of the car decreases. Where does this energy go? Where does most of it go in a hybrid vehicle?48. What is the theoretical mechanical advantage for each of the three lever systems shown?47. Energy is wasted as heat in a non-hybrid car. In a hybrid car, energy charges batteries and is converted to electricity.48. 1 2 0.5
49. Dry-rock geothermal power can be a major contributor to power with no pollution. The bottom of a hole drilled down into Earthโs interior is fractured, making a large surfaced hot cavity. Water is introduced from the top by a second hole. Superheated water rising to the surface then drives a conventional turbine to produce electricity. What is the source of this energy?50. A stuntman on a cliff has a PE of 10,000 J. Show that when his potential energy is 2000 J, his kinetic energy is 8000 J.49. Energy from radioactive decay in Earthโs interior50. KE =
51. Relative to the ground below, how many joules of PE does a 1000-N boulder have at the top of a 5-m ledge? If it falls, with how much KE will it strike the ground? What will be its speed on impact?
52. A hammer falls off a rooftop and strikes the ground with a certain KE. If it fell from a roof that was four times higher, how would its KE of impact compare? Its speed of impact? (Neglect air resistance.)Four times higher means four times the PE, thus four times the KE of impact. From
This means twice the impact speed (because ). This result can be obtained from
โฆwhere falling from 4d takes twice the time. Twice the time at the same acceleration g means twice the speed; .
53. A car can go from 0 to 100 km/h in 10 s. If the engine delivered twice the power, how many seconds would it take?54. If a car traveling at 60 km/h will skid 20 m when its brakes lock, how far will it skid if it is traveling at 120 km/h when its brakes lock? (This question is typical on some driverโs license exams.)53. Twice the power means doing twice the work in the same time or the same work in half the time. To achieve the same change in speed (and same change in KE) with twice the power means the work can be done in half the time, or 5 s.54. Twice the speed means four times the KE, and four times the work to reduce the KE to zero. F is constant so d = 80 m.
1. A skater with a mass of 52.0 kg moving at 2.5 m/s glides to a stop over a distance of 24.0 m. How much work did the friction of the ice do to bring the skater to a stop? How much work would the skater have to do to speed up to 2.5 m/s again? ๐=๐ฒ๐ฌ ๐ โ๐ฒ๐ฌ๐
๐=+๐๐๐ ๐ฑ
๐=๐๐๐ฆ๐ ๐
๐โ๐๐๐ฆ๐๐
๐
๐=๐๐
(๐๐๐ค๐ )(๐ ๐๐ )
๐
โ๐๐
(๐๐๐ค๐ )(๐ .๐๐๐ )
๐
2. An 875.0-kg compact car speeds up from 22.0 m/s to 44.0 m/s while passing another car. What are its initial and final energies, and how much work is done on the car to increase its speed? ๐=๐ฒ๐ฌ ๐ โ๐ฒ๐ฌ๐
๐=+๐๐๐ ,๐๐๐ ๐ฑ
๐=๐๐๐ฆ๐ ๐
๐โ๐๐๐ฆ๐๐
๐
๐=๐๐
(๐๐๐๐ค๐ )(๐๐ ๐๐ )
๐
โ๐๐
(๐๐๐๐ค๐ )(๐๐๐๐ )
๐
3. A comet with a mass of 7.85 x1011 kg strikes Earth at a speed of 25.0 km/s. Find kinetic energy of the comet in joules, and compare the work that is done by Earth in stopping the comet to the 4.2x 1015 J of energy that was released by the largest nuclear weapon ever built.
๐ฒ๐ฌ ๐๐๐๐๐=๐ .๐๐๐ฑ๐๐๐๐ ๐ฑ
๐=๐๐๐ฆ๐๐
๐ฒ๐ฌ ๐๐๐๐๐=๐๐
(๐ .๐๐๐ฑ๐๐๐๐๐ค๐ )(๐ .๐๐ฑ๐๐๐ ๐๐ )
๐
5.8 x 104 bombs would be required to produce the same amount of energy used by Earth in stopping the comet.
๐ฒ๐ฌ๐๐๐๐๐
๐ฒ๐ฌ๐๐๐๐
=๐ .๐๐๐ฑ๐๐๐๐ ๐ฑ ๐ .๐๐ฑ ๐๐๐๐ ๐ฑ
=๐ .๐๐ฑ๐๐๐ ๐ฑ
4. In Example Problem 1, what is the potential energy of the bowling ball relative to the rack when it is on the floor?
PE =mgh PE = (7.30 kg)(9.80 m/s2)(-0.610 m)
PE =-43.6 J5. If you slowly lower a 20.0 kg bag of sand 1.20 m from the trunk of a car to the driveway, how much work do you do?
W = Fd W = mg(hf hi)
W = (20.0 kg)(9.80 m/s2)(0.00 m - 1.20 m)
W = 2.35x102 J
6. A boy lifts a 2.2-kg book from his desk, which is 0.80 m high, to a bookshelf that is 2.10 m high. What is the potential energy of the book relative to the desk?
7. If a 1.8-kg brick falls to the ground from a chimney that is 6.7 m high, what is the change in its potential energy?
๐ท๐ฌ=๐๐ (๐ ๐โ๐ ๐ )๐ท๐ฌ=(๐ .๐๐๐ )(๐ .๐๐
๐๐) (๐ .๐๐โ๐.๐๐ฆ )๐ท๐ฌ=๐๐ ๐ฑ
๐ท๐ฌ=๐๐ (๐ ๐โ๐ ๐ )๐ท๐ฌ=๐๐ (๐ ๐โ๐ ๐ )๐ท๐ฌ=(๐ .๐๐๐ )(๐ .๐๐
๐๐ )(๐๐โ๐ .๐๐ฆ )
๐ท๐ฌ=โ๐๐๐ ๐ฑ
8. A warehouse worker picks up a 10.1-kg box from the floor and sets it on a long, 1.1-m-high table. He slides the box 5.0 m along the table and then lowers it back to the floor. What were the changes in the energy of the box, and how did the total energy of the box change? (Ignore friction.)
To slide the box across the table, W 0.0 because the height did not change and we ignored friction. To lower the box to the floor:
โ ๐ท๐ฌ= (๐๐ .๐๐๐ )(๐ .๐๐๐๐ ) (๐.๐๐โ๐๐ฆ )=๐๐๐ ๐
๐ท๐ฌ=(๐๐ .๐๐๐ )(๐ .๐๐๐๐) (๐๐โ๐ .๐๐ฆ )=โ๐๐๐ ๐
๐พ =๐ญ๐ =๐ท๐ฌ=๐๐ (๐๐ โ๐๐ )=โ๐ท๐ฌ
๐พ =๐ญ๐ =๐ท๐ฌ=๐๐ (๐๐ โ๐๐ )=โ๐ท๐ฌ
8. A warehouse worker picks up a 10.1-kg box from the floor and sets it on a long, 1.1-m-high table. He slides the box 5.0 m along the table and then lowers it back to the floor. What were the changes in the energy of the box, and how did the total energy of the box change? (Ignore friction.)
๐ท๐ฌ=(๐๐ .๐๐๐ )(๐ .๐๐๐๐) (๐๐โ๐ .๐๐ฆ )=โ๐๐๐ ๐
๐พ =๐ญ๐ =๐ท๐ฌ=๐๐ (๐๐ โ๐๐ )=โ๐ท๐ฌ
The sum of the three energy changes isโฆ๐๐๐ ๐ฑ+๐ ๐ฑ +(โ๐๐๐ ๐ฑ )=๐ ๐ฑ
9. Elastic Potential Energy You get a spring loaded toy pistol ready to fire by compressing the spring. The elastic potential energy of the spring pushes the rubber dart out of the pistol. You use the toy pistol to shoot the dart straight up. Draw bar graphs that describe the forms of energy present in the following instances. a. The dart is pushed into the gun barrel, thereby compressing the spring.There should be three bars:
one for the springโs potential energy, one for gravitational potential energy, and one for kinetic energy. The springโs potential energy is at the maximum level, and the other two are zero.
b. The spring expands and the dart leaves the gun barrel after the trigger is pulled.
The kinetic energy is at the maximum level, and the other two are zero.
c. The dart reaches the top of its flight.
The gravitational potential energy is at the maximum level, and the other two are zero.
10. Potential Energy A 25.0-kg shell is shot from a cannon at Earthโs surface. The reference level is Earthโs surface. What is the gravitational potential energy of the system when the shell is at 425 m? What is the change in potential energy when the shell falls to a height of 225 m?
๐ท๐ฌ=(๐๐๐๐ )(๐ .๐ ๐๐๐ ) (๐๐๐๐ )=๐๐๐ ,๐๐๐ ๐
๐๐๐ ,๐๐๐ ๐ฑ โ๐๐ ,๐๐๐ ๐ฑ=๐๐ ,๐๐๐ ๐ฑ
๐ยฟ๐ท๐ฌ=๐๐๐๐ยฟ ๐ท๐ฌ=๐๐๐
๐ท๐ฌ=(๐๐๐๐ )(๐ .๐ ๐๐๐ ) (๐๐๐๐ )=๐๐ ,๐๐๐ ๐
The change in energy isโฆ
11. Rotational Kinetic Energy Suppose some children push a merry-go-round so that it turns twice as fast as it did before they pushed it. What are the relative changes in angular momentum and rotational kinetic energy?The angular momentum is doubled because it is proportional to the angular velocity. The rotational kinetic energy is quadrupled because it is proportional to the square of the angular velocity. The children did work in rotating the merry-go-round.
12. Work-Energy Theorem How can you apply the work-energy theorem to lifting a bowling ball from a storage rack to your shoulder?The bowling ball has zero kinetic energy when it is resting on the rack or when it is held near your shoulder. Therefore, the total work done on the ball by you and by gravity must equal zero.
13. Potential Energy A 90.0-kg rock climber first climbs 45.0 m up to the top of a quarry, then descends 85.0 m from the top to the bottom of the quarry. If the initial height is the reference level, find the potential energy of the system (the climber and Earth) at the top and at the bottom. Draw bar graphs for both situations.
๐จ๐๐ป๐๐ ๐ท๐ฌ=(๐๐๐๐ )(๐ .๐๐๐๐) (+๐๐๐ )=๐๐ ,๐๐๐ ๐
๐ท๐ฌ=๐๐๐๐จ๐๐ป๐๐ ๐ท๐ฌ=๐๐ ,๐๐๐ ๐
๐จ๐ ๐ฉ๐๐๐๐๐๐ท๐ฌ=(๐๐๐๐)(๐.๐ ๐๐๐ ) (โ๐๐๐ )=๐๐ ,๐๐๐ ๐
14. Karl uses an air hose to exert a constant horizontal force on a puck, which is on a frictionless air table. He keeps the hose aimed at the puck, thereby creating a constant force as the puck moves a fixed distance.a. Explain what happens in terms of work and energy. Draw bar graphs. Karl exerted a constant force F over a distance d and did an amount of work W =Fd on the puck. This work changed the kinetic energy of the puck.
b. Suppose Karl uses a different puck with half the mass of the first one. All other conditions remain the same. How will the kinetic energy and work differ from those in the first situation?If the puck has half the mass, it still receives the same amount of work and has the same change in kinetic energy. However, the smaller mass will move faster by a factor of 1.414.
๐=๐ฒ๐ฌ ๐ โ๐ฒ๐ฌ๐๐=๐๐๐ฆ๐ ๐
๐โ๐๐๐ฆ๐๐
๐๐=๐๐๐ฆ๐ ๐
๐
c. Explain what happened in parts a and b in terms of impulse and momentum. The two pucks do not have the same final momentum. Momentum of the first puck:
P1 = m1 + v1
Momentum of the second puck:P2 = m2v2
Thus, the second puck has less momentum than the first puck does. Because the change in momentum is equal to the impulse provided by the air hose, the second puck receives a smaller impulse.
๐=(๐๐๐๐)(๐ .๐๐๐๐ยฟยฟ๐)= .๐๐๐๐๐ยฟ
15. A bike rider approaches a hill at a speed of 8.5 m/s. The combined mass of the bike and the rider is 85.0 kg. Choose a suitable system. Find the initial kinetic energy of the system. The rider coasts up the hill. Assuming there is no friction, at what height will the bike come to rest?The system is the (bike + rider + Earth). There are no external forces, so total energy is conserved.
๐ฒ๐ฌ ๐+๐ท๐ฌ๐=๐ฒ๐ฌ ๐ +๐ท๐ฌ ๐
๐๐=๐๐๐ฆ ๐๐๐๐=๐
๐(๐๐๐ค๐ )(๐ .๐ ๐
๐ )๐๐๐=๐ ,๐๐๐ ๐ฑ
๐๐๐ฆ ๐๐+๐=๐+๐ฆ๐ ๐ก
๐= ๐๐
๐๐๐=
(๐ .๐๐๐ )
๐
๐(๐ .๐ ๐๐๐ )
๐=๐.๐๐
16. Suppose that the bike rider in problem 15 pedaled up the hill and never came to a stop. In what system is energy conserved? From what form of energy did the bike gain mechanical energy?The system of Earth, bike, and rider remains the same, but now the energy involved is not mechanical energy alone. The rider must be considered as having stored energy, some of which is converted to mechanical energy. Energy came from the chemical potential energy stored in the riderโs body.
17. A skier starts from rest at the top of a 45.0-m-high hill, skis down a 30ยฐ incline into a valley, and continues up a 40.0-mhigh hill. The heights of both hills are measured from the valley floor. Assume that you can neglect friction and the effect of the ski poles. How fast is the skier moving at the bottom of the valley? What is the skierโs speed at the top of the next hill? Do the angles of the hills affect your answers?
๐ฒ๐ฌ ๐+๐ท๐ฌ๐=๐ฒ๐ฌ ๐ +๐ท๐ฌ ๐๐ฉ๐๐๐๐๐
๐+๐ฆ๐ ๐ก=๐๐๐ฆ ๐๐+๐
๐๐=๐๐๐
๐=๐๐ .๐๐๐
๐=โ๐๐๐๐=โ๐ (๐ .๐ ๐
๐๐ )(๐๐๐)
๐ฒ๐ฌ ๐+๐ท๐ฌ๐=๐ฒ๐ฌ ๐ +๐ท๐ฌ ๐
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๐+๐ฆ๐ ๐๐=๐๐๐ฆ๐๐+๐ฆ๐ ๐ ๐๐๐=๐๐ (๐ยฟยฟ๐โ๐ ๐ )ยฟ
๐=๐ .๐๐๐ ๐=โ๐๐(๐ยฟยฟ๐โ๐ ๐ ) ยฟ
๐=โ๐ (๐ .๐ ๐๐๐ )(๐๐๐โ๐๐๐)
No, the angles do not have any impact.
18. In a belly-flop diving contest, the winner is the diver who makes the biggest splash upon hitting the water. The size of the splash depends not only on the diverโs style, but also on the amount of kinetic energy that the diver has. Consider a contest in which each diver jumps from a 3.00-m platform. One diver has a mass of 136 kg and simply steps off the platform. Another diver has a mass of 102 kg and leaps upward from the platform. How high would the second diver have to leap to make a competitive splash?Using the water as a reference level, the kinetic energy on entry is equal to the potential energy of the diver at the top of his flight.
Using the water as a reference level, the kinetic energy on entry is equal to the potential energy of the diver at the top of his flight. The large diver hasโฆ ๐๐=๐ฆ๐ ๐ก
๐ท๐ฌ=(๐๐๐๐๐)(๐ .๐๐๐๐ ) (๐๐ )=๐๐๐๐ ๐ฑ
๐=๐๐๐๐ ๐ฑ
(๐๐๐๐๐)(๐ .๐๐๐๐ )
=๐๐ฆ
19. An 8.00-g bullet is fired horizontally into a 9.00-kg block of wood on an air table and is embedded in it. After the collision, the block and bullet slide along the frictionless surface together with a speed of 10.0 cm/s. What was the initial speed of the bullet?
๐๐=(๐ฆ+๐ )๐๐๐
๐=(๐ฆ+๐ )๐
๐
๐=(๐ .๐๐๐๐๐ค๐ +๐๐ค๐ )(๐ .๐ ๐
๐ )
๐ .๐๐๐๐๐๐
๐=๐๐๐๐๐
20. A 0.73-kg magnetic target is suspended on a string. A 0.025-kg magnetic dart, shot horizontally, strikes the target head-on. The dart and the target together, acting like a pendulum, swing 12.0 cm above the initial level before instantaneously coming to rest.a. Sketch the situation and choose a system.
The system includes the suspended target and the dart.
b. Decide what is conserved in each part and explain your decision. Only momentum is conserved in the inelastic dart-target collision, so
mvo + MVo = (m + M)Vf
Where, Vi = 0 since the target is initially at rest and Vf is the common velocity just after impact. As the dart-target combination swings upward, energy is conserved, so
ฮPE = ฮKE or, at the top of the swing,
(๐ฆ+๐ )๐ ๐ ๐=๐๐
(๐ฆ+๐) (๐ฝ ๐ )๐
b. What was the initial velocity of the dart?Solve for Vf โฆ
๐ฝ ๐=๐๐๐๐
๐ฝ ๐=โ๐๐๐๐
Substitute vf into the momentum equation and solve for vi.
๐ฝ ๐=(๐ .๐๐๐๐๐+๐ .๐๐๐๐๐ .๐๐๐๐๐ )โ๐ (๐ .๐ ๐
๐๐ )(๐ .๐๐๐ )
b. What was the initial velocity of the dart?Solve for Vf โฆ
๐ฝ ๐=๐๐๐๐
๐ฝ ๐=โ๐๐๐๐
Substitute vf into the momentum equation and solve for vi.
๐ฝ ๐=(๐ .๐๐๐๐๐+๐ .๐๐๐๐๐ .๐๐๐๐๐ )โ๐ (๐ .๐ ๐
๐๐ )(๐ .๐๐๐ )
21. A 91.0 kg hockey player is skating on ice at 5.50 m/s. Another hockey player of equal mass, moving at 8.1 m/s in the same direction, hits him from behind. They slide off together.
a. What are the total energy and momentum in the system before the collision?
๐ฒ๐ฌ ๐=๐๐
๐๐(๐ ยฟยฟ๐)๐+๐๐
๐๐(๐ยฟยฟ๐)๐ ยฟยฟ
๐ฒ๐ฌ ๐=๐๐๐๐๐๐ (๐ .๐๐
๐ )๐
+๐๐๐๐๐๐(๐ .๐ ๐
๐ )๐
๐ท ๐=๐๐๐๐+๐๐๐๐
๐ท ๐=(๐๐๐๐ )(๐ .๐๐๐ )+(๐๐๐๐ )(๐ .๐๐
๐ )๐ท ๐=(๐ ,๐๐๐ ๐๐๐
๐ )
๐ฒ๐ฌ ๐=๐ ,๐๐๐ ๐ฑ
21. A 91.0 kg hockey player is skating on ice at 5.50 m/s. Another hockey player of equal mass, moving at 8.1 m/s in the same direction, hits him from behind. They slide off together.
b. What is the velocity of the two hockey players after the collision?
๐ท ๐=๐๐๐๐+๐๐๐๐ ๐ ๐=๐.๐๐๐
๐๐๐๐+๐๐๐๐=ยฟ ยฟ
๐ท ๐=๐ท ๐
๐ ๐=๐๐๐๐+๐๐๐๐
๐๐+๐๐
๐ ๐=(๐๐๐๐)(๐ .๐ ๐
๐ )+(๐๐๐๐)(๐.๐๐๐ )
๐๐๐๐๐
21. A 91.0 kg hockey player is skating on ice at 5.50 m/s. Another hockey player of equal mass, moving at 8.1 m/s in the same direction, hits him from behind. They slide off together.c. How much energy was lost in the collision?The KEf is:
๐ฒ๐ฌ ๐=๐๐
ยฟ ยฟ
๐ฒ๐ฌ ๐=๐๐
(๐๐๐๐+๐๐๐๐)(๐ .๐ ๐๐ )
๐
๐ฒ๐ฌ ๐=๐ ,๐๐๐ ๐ฑโ ๐ฒ๐ฌ=๐ฒ๐ฌ ๐ โ๐ฒ๐ฌ ๐=๐ ,๐๐๐ ๐ฑ โ๐ ,๐๐๐ ๐ฑ
โ ๐ฒ๐ฌ=๐๐๐ ๐ฑ