Mechanics Problems - Conservation of Energy and Conservation of Momentum

Conservation of Energy (Mechanical, Gravitational)

  1. You are watching a National Geographic Special on television. One segment of the program is about archer fish, which inhabit streams in southeast Asia. This fish actually "shoots" water at insects to knock them into the water so it can eat them. The commentator states that the archer fish keeps its mouth at the surface of the stream and squirts a jet of water from its mouth at 13 feet/second. You watch an archer fish shoot a juicy moth off a leaf into the water. You estimate that the leaf was about 2.5 feet above a stream. You wonder at what minimum angle from the horizontal the water can be ejected from the fish's mouth to hit the moth. Since you have time during the commercial, you quickly calculate this angle.

  2. Your artist friend is designing a kinetic sculpture and asks for your help since she knows that you have had physics. Part of her sculpture consists of a 6.0-kg object (you can't tell what it is supposed to be, but it's art) and a 4.0-kg object which hang straight down from opposite ends of a very thin, flexible wire. This wire passes over a smooth, cylindrical, horizontal, stainless steel pipe 3.0 meters above the floor. The frictional force between the rod and the wire is negligible. The 6.0-kg object is held 2.0 meters above the floor and the other object hangs 0.50 meters above the floor. When the mechanism releases the 6.0-kg object, both objects accelerate and one will eventually hit the floor -- but they don't hit each other. To determine if the floor will be damaged, calculate the speed of the object which hits the floor.

  3. You are driving your car uphill along a straight road. Suddenly, you see a car run a red light and enter the intersection just ahead of you. You slam on your brakes and skid in a straight line to a stop, leaving skid marks 100 feet long. A policeman observes the whole incident and gives a ticket to the other car for running a red light. He also gives you a ticket for exceeding the speed limit of 30 mph. When you get home, you read your physics book and estimate that the coefficient of kinetic friction between your tires and the road was 0.60, and the coefficient of static friction was 0.80. You estimate that the hill made an angle of about 10owith the horizontal. You look in your owner's manual and find that your car weighs 2,050 lbs. Will you fight the traffic ticket in court?

  4. You have landed a summer job with a company that has been given the contract to design the ski jump for the next Winter Olympics. The track is coated with snow and has an angle of 25o from the horizontal. A skier zips down the ski jump ramp so that he leaves it at high speed. The winner is the person who jumps the farthest after leaving the end of the ramp. Your task is to determine the height of the starting gate above the end of the ramp, which will determine the mechanical structure of the ski jump facility. You have been told that the typical ski-jumper pushes off from the starting gate at a speed of 2.0 m/s. For safety reasons, your design should be such that for a perfect run down the ramp, the skier's speed before leaving the end of the ramp and sailing through the air should be no more than 80 km/hr. You run some experiments on various skies used by the jumpers and determine that the coefficient of static friction between the snow and the skis is 0.10 and its coefficient of kinetic friction is 0.02. Since the ski-jumpers bend over and wear very aerodynamic suits, you decide to neglect the air resistance to make your design.

  5. The Navy wants a new airplane launcher for their aircraft carriers that is basically a large spring and your job is to determine the necessary spring constant. The launcher pushes the plane for a short distance along a much longer runway. During that same time, the plane's jet engines supply a constant thrust force for the entire length of the runway. The planes need to have a minimum velocity by the time they reach the end of the runway in order to take off successfully.

  6. You have been hired to design a safety system to protect drivers going down hills during an ice storm. The planned system consists of a bumper, which can be considered a stiff spring, at the bottom of the hill. In the scenario you are given, the car starts from rest at the top of a hill which makes an angle q with the horizontal. The distance that the car slides from the top of the hill until it is stopped by the spring is L. For the worst case scenario, assume that there is no frictional force between the car and road due to the ice. If the maximum compression of the spring from its equilibrium position is D, your job is to calculate the required spring constant k in terms of D, L and q.

  7. You work for the National Park Service testing a small cannon used to prevent avalanches by shooting down snow overhanging the sides of mountains. In order to determine the range of the cannon, it is necessary to know the speed with which the projectile leaves the cannon (muzzle speed), relative to the ground. The cannon you are testing has a weight of 500 lbs. and shoots a 20-lb. projectile. During lab tests where the cannon is held and cannot move, the muzzle speed is 400 m/s. You want to calculate the projectile's muzzle speed with respect to the ground under field conditions when the cannon is mounted so that it is free to move (recoil) when fired. You take the case where the cannon is fired horizontally using the same shells as in the laboratory.

  8. Super Dave has just returned from the hospital where he spent a week convalescing from injuries incurred when he was "shot" out of a cannon to land in an airbag which was too thin. Undaunted, he decides to celebrate his return with a new stunt. He intends to jump off a 100-foot tall tower with an elastic cord tied to one ankle, and the other end tied to the top of the tower. This cord is very light but very strong and stretches so that it can stop him without pulling his leg off. Such a cord exerts a force with the same mathematical form as the spring force. He wants it to be 75 feet long so that he will be in free fall for 75 feet before the cord begins to stretch. To minimize the force that the cord exerts on his leg, he wants it to stretch as far as possible. You have been assigned to purchase the cord for the stunt and must determine the elastic force constant which characterizes the cord that you should order. Before the calculation, you carefully measure Dave's height to be 6.0 ft and his weight to be 170 lbs. For maximum dramatic effect, his jump will be off a diving board at the top of the tower. From tests you have made, you determine that his maximum speed coming off the diving board is 10 ft/sec. Neglect air resistance in your calculation -- let Dave worry about that.

  9. As part of a fundraiser, you want the new dean to bungee jump from a crane. The jump will be made from 44 m above a 2.5 m deep pool of Jello. A 30 m long bungee cord would be attached to the dean's ankle. You must convince the dean that your plan is safe for a person of his mass, 70 kg. As the bungee cord stretches, it will exert a force with the same properties as the force exerted by a spring. Your plan has the dean stepping off a platform and being in free fall for the 30 m before the cord begins to stretch. You must determine the elastic constant of the bungee cord so that it stretches only 12 m, which will keep the dean's head just out of the Jello.

  10. Your artist friend's new work is a simple, high-impact kinetic sculpture called 'Destruction.' A 200-kg steel block is hung from the ceiling by an 8-foot-long rope. A second rope is attached to the side of the block. The other end of this second rope is attached to a motor which is cleverly mounted so that the rope always pulls the block horizontally with a constant force. The block starts from rest, hanging straight down, and is pulled slowly by the motor until it is hanging at an angle of 30 degrees from the vertical. The horizontal rope is then released and the block swings and crashes into a wall. Your friend knows you have taken physics and asks you the minimum energy that the motor must supply. You perform a test and determine that the block is in equilibrium when it has been pulled so that it hangs at 30 degrees from the vertical.

  11. (Gravitational Energy) Because of your knowledge of physics and interest in the environment, you have gotten a summer job with an organization which wants to orbit a satellite to monitor the amount of chlorine ions in the upper atmosphere over North America. It has been determined that the satellite should collect samples at a height of 100 miles above the Earth's surface. Unfortunately, at that height air resistance would make the amount of time the satellite would stay in orbit too short to be useful. You suggest that an elliptical orbit would allow the satellite to be close to the Earth over North America, where data was desired, but farther from the Earth, and thus out of almost all of the atmosphere, on the other side of our planet. Your colleague estimates that the satellite would be traveling at 10,000 miles/hour when it was farthest from the Earth at a height of 1,000 miles. How fast would the satellite be traveling when it took its air samples if you neglect air friction?


Conservation of Energy (Mechanical) and Force

  1. At the train station, you notice a large horizontal spring at the end of the track where the train comes in. This is a safety device to stop the train so that it will not plow through the station if the engineer misjudges the stopping distance. While waiting, you wonder what would be the fastest train that the spring could stop at its full compression, 3.0 ft. To keep the passengers safe when the train stops, you assume a maximum stopping acceleration of g/2. You also guess that a train weighs half a million lbs. For purpose of getting an estimate, you decide to assume that all frictional force are negligible.

  2. Your company is designing an apparatus for an ice skating show. An ice skater will start from rest and slide down an ice-covered ramp. At the bottom of the ramp, the skater will glide around an ice-covered loop which is the inside of a vertical circle before emerging out onto the skating rink floor. For a spectacular effect, the circular loop will have a diameter of 30 feet. Your task is to determine the minimum height from the rink floor to the top of the ramp for the skater to make it around the loop. When barely making it around, the skater briefly loses contact with the ice at the top of the loop and at that point the skater is in free fall.

  3. In a weak moment you have volunteered to be a human cannonball at an amateur charity circus. The "cannon" is actually a 3-foot diameter tube with a big stiff spring inside which is attached to the bottom of the tube. A small seat is attached to the free end of the spring. The ringmaster, one of your soon to be ex-friends, gives you your instructions. He tells you that just before you enter the mouth of the cannon, a motor will compress the spring to 1/10 its normal length and hold it in that position. You are to gracefully crawl in the tube and sit calmly in the seat without holding on to anything. The cannon will then be raised to an angle such that your speed through the air at your highest point is 10 ft/sec. When the spring is released, neither the spring nor the chair will touch the sides of the 12-foot long tube. After the drum roll, the spring is released and you will fly through the air with the appropriate sound effects and smoke. With the perfect aim of your gun crew, you will fly through the air over a 15-foot wall and land safely in the net. You are just a bit worried and decide to calculate how high above your starting position you will be at your highest point. Before the rehearsal, the cannon is taken apart for maintenance. You see the spring, which is now removed from the cannon, is hanging straight down with one end attached to the ceiling. You determine that it is 10 feet long. When you hang on its free end without touching the ground, it stretches by 2.0 ft. Is it possible for you to make it over the wall?


Conservation of Momentum

  1. You are on a committee investigating injuries to students participating in sports, starting with the high incidence of ankle injuries on the basketball team. Observing the team practice jump shots inspires you to try a small calculation. A 50-kg student jumps 1.0 meter straight up and shoots the 0.80-kg basketball at the top of the jump. From the path of the basketball, you estimate that the ball left the hand at 30 degrees from the horizontal at 5 m/s. To determine the horizontal forces on the ankle, you decide to calculate the student's horizontal velocity when hitting the ground.

  2. Because of your interest in the environment and your physics experience, you have been asked by the Campus Museum of Natural History to assist in the production of an animated film about hawks. In the script, a 1.5-kg hawk hovers motionless with respect to the ground when it sees a goose flying below it. The hawk dives straight down. It strikes the goose at a speed of 60 km/hr and digs its claws into the goose's body. The 2.5 kg goose was flying north at 30 km/hr just before it was struck by the hawk and killed instantly. The animators want to know the velocity of the hawk and dead goose just after the strike.

  3. As part of an interview for a summer job with the Coast Guard, you are asked to help determine the search area for two sunken ships by calculating their velocity just after they collided. According to the last radio transmission from the 40,000-ton luxury liner, the Hedonist, it was going due west at a speed of 20 knots in calm seas through a rare fog just before it was struck broadside by the 60,000-ton freighter, the Ironhorse, which was traveling north at 10 knots. The transmission also noted that when the freighter's bow pierced the hull of the liner, the two ships stuck together and sank together.

  4. You have been hired to check the technical correctness of an upcoming made-for-TV murder mystery that takes place in the space shuttle. In one scene, an astronaut's safety line is cut while on a space walk. The astronaut, who is 200 meters from the shuttle and not moving with respect to it, finds that the suit's thruster pack has also been damaged and no longer works and that only 4 minutes of air remains. To get back to the shuttle, the astronaut unstraps a 10-kg tool kit and throws it away with a speed of 8 m/s. In the script, the astronaut, who has a mass of 80 kg without the toolkit, survives, but is this correct?


Conservation of Energy (Mechanical) and Momentum

  1. You have been hired as a technical consultant for an early-morning cartoon series for children to make sure that the science is correct. In the script, a wagon containing two boxes of gold (total mass of 150 kg) has been cut loose from the horses by an outlaw. The wagon starts from rest 50 meters up a hill with a 6o slope. The outlaw plans to have the wagon roll down the hill and across the level ground and then crash into a canyon where his confederates wait. But in a tree 40 meters from the edge of the canyon wait the Lone Ranger (mass 80 kg) and Tonto (mass 70 kg). They drop vertically into the wagon as it passes beneath them. The script states that it takes the Lone Ranger and Tonto 5 seconds to grab the gold and jump out of the wagon, but is this correct?. You assume that the wagon rolls with negligible friction.

  2. You are helping your friend prepare for the next skateboard exhibition by determining if the planned program will work. Your friend will take a running start and then jump onto a heavy-duty 15-lb stationary skateboard. The skateboard will glide in a straight line along a short, level section of track, then up a sloped concrete wall. The goal is to reach a height of at least 10 feet above the starting point before coming back down the slope. Your friend's maximum running speed to safely jump on the skateboard is 23 feet/second. Your friend weighs 150 lbs.

  3. Because of your physics background, you have been hired as a technical advisor for a new James Bond adventure movie. In the script, Bond and his latest love interest, who is 2/3 his weight (including skis, boots, clothes, and various hidden weapons), are skiing in the Swiss Alps. She skis down a slope while he stays at the top to adjust his boot. When she has skied down a vertical distance of 100 ft, she stops to wait for him and is captured by the bad guys. Bond looks up and sees what is happening. He notices that she is standing with her skis pointed downhill while she rests on her poles. To make as little noise as possible, Bond starts from rest and glides down the slope heading right at her. Just before they collide, she sees him coming and lets go of her poles. He grabs her and they both continue downhill together. At the bottom of the hill, another slope goes uphill and they continue to glide up that slope until they reach the top of the hill and are safe. The writers want you to calculate the maximum possible height that the second hill can be relative to the position where the collision took place. Both Bond and his girl friend are using new, top-secret frictionless stealth skis developed for the British Secret Service.

  4. Because of your concern that incorrect science is being taught to children when they watch cartoons on TV, you have joined a committee which is reviewing a new cartoon version of Tarzan. In this episode, Tarzan is on the ground in front of a herd of stampeding elephants. Just in time Jane, who is up in a tall tree, sees him. She grabs a convenient vine and swings towards Tarzan, who has twice her mass, to save him. Luckily, the lowest point of her swing is just where Tarzan is standing. When she reaches him, he grabs her and the vine. They both continue to swing to safety over the elephants up to a height which looks to be about 1/2 that of Jane's original position. To decide if you going to approve this cartoon, calculate the maximum height Tarzan and Jane can swing as a fraction of her initial height.

  5. You are watching a Saturday morning cartoon concerning a jungle hero called George of the Jungle. George attempts to save his friend, an ape named Ape, from a stampeding herd of wildebeests. Ape is at the base of a tall tree which has a vine attached to its top. George is in another tree holding the other end of the vine. George plans to swing down from the tree, grab Ape at the bottom of the swing, and continue up to safety on a ledge which is half of George's initial height in the tree. Assuming that Ape weighs the same as George, will they successfully make it to the top of the ledge?

  6. Your friend has just been in a traffic accident and hopes that you can show the accident was the other drivers fault. Your friends car was traveling North when it entered the intersection. When it reached the center of the intersection, the car was struck by the other drivers car which was traveling East. The two cars remained joined together after the collision and skidded to a stop. The speed limit on both roads is 50 mph. From the skid marks still visible on the street, you determine that after the collision the cars skidded 56 feet at an angle of 30 degrees north of east before stopping. The police report gives the make and year of each car. The weight of your friends car is 2600 lbs and that of the other car is 2200 lbs, including the drivers weight in each case. The coefficient of kinetic friction for a rubber tire skidding on dry pavement is 0.80. You decide to see if the other driver was speeding and if your friend was under the speed limit.

  7. Because movie producers have come under pressure for teaching children incorrect science, you have been appointed to help a committee of concerned parents review a script for a new Superman movie. In the scene under consideration, Superman rushes to save Lois Lane who has been pushed from a window 300 feet above a crowded street. Superman is 0.5 miles away when he hears Lois scream and rushes to save her. He swoops down in the nick of time, arriving when Lois is just 3.0 feet above the street, and stopping her just at ground level. Lois changes her expression from one of horror at her impending doom to a smile of gratitude as she gently floats to the ground in Superman's arms. The committee wants to know if there is really enough time to express this range of emotions, even if there is a possible academy award on the line. The chairman asks you to calculate the time it takes for Superman to stop Lois's fall. To do the calculation, you assume that Superman applies a constant force to Lois in breaking her fall and that she weighs 120 lbs. While thinking about this scene you also wonder if Lois could survive the force that Superman applies to her.

  8. This year you have a summer job working for the National Park Service. Since they know that you have taken physics, they start you off in the laboratory which tests possible new equipment. Your first job is to test a small cannon. During the winter, small cannons are used to prevent avalanches in populated areas by shooting down heavy snow concentrations overhanging the sides of mountains. In order to determine the range of the cannon, it is necessary to know the velocity with which the projectile leaves the cannon (muzzle velocity). The cannon you are testing has a weight of 700 lbs and shoots a 40-lb projectile. During the lab tests the cannon is held horizontally in a rigid support so that it cannot move. Under those conditions, you measure the magnitude of the muzzle velocity to be 400 m/s. When the cannon is actually used in the field, however, it is mounted so that it is free to move (recoil) when it is fired. Your boss asks you to calculate the projectile's speed leaving the cannon under field conditions, when it is allowed to recoil. She tells you to take the case where the cannon is fired horizontally using cannon shells which are identical to those used in the laboratory test.

  9. For a part time job with a medical physics group, you are studying a cancer therapy that uses neutrons to knock a particle out of the nucleus of the atoms of cancer cells. This is an inelastic collision in which the heavy nucleus essentially does not move. After the collision, the nucleus decays and kills the cancer cell. You decide to measure the change in internal energy of a nitrogen nucleus after a neutron collides with it. In the experiment, a neutron hits the nucleus with a speed of 2.0 x 10^7 m/s and you detect two neutrons both coming out at angles of 30 degrees with respect to the direction that the neutron coming in.