Chapter 3
Suppose a gliding cart with a mass of 0.5 kg bumps into, and sticks to, a stationary cart that has a mass of 1.5 kg. IF the speed of the gliding cart before impact is V(before), how fast will the coupled carts glide after the collision?
According to momentum conservation, the momentum of the 0.5-kg cart before the collision = the momentum of both carts stuck together afterward. (.0.5 kg) v-before = (0.5kg + 1.5kg) v-after. V-after = 0.5kg V-before ÷ (0.5kg + 1.5kg) = 0.5kg v-before ÷ 2 kg = v-before ÷ 4. This makes sense, because four times as much mass will be moving after the collision, so the coupled carts will glide more slowly. The same momentum means that four times the mass glades 1/4 as fast.
Assuming you have average strength, can you lift a 160-kg object with your bare hands? Can you do 1600 J of work on it?
An object with a mass of 160 kg weighs 1600 N, or 352 lbs. So no, you cannot lift it without the use of some type of machine. If you can't move it, you can't do work on it. You'd do 1600 J of work on it if you could lift it a vertical distance of 1m.
For the same force, why does a longer cannon impart more speed to a cannonball?
As learned earlier, a longer barrel imparts more impulse because of the longer time during which the force acts. The work-energy theorem similarly tells us that the longer the distance over which the force acts, the greater the change in kinetic energy. So we see two reason for cannons with long barrels producing greater cannonball speeds.
IF you throw a raw egg against a wall, it will break; but if you throw it withe same speed into a sagging sheet, the egg won't break. Why?
Because the sheet allows the egg to have a smaller force upon impact than the it would against the wall.
Compare the momentum of a 1-kg cart moving at 10 m/s with that of a 2-kg cart moving at 5 m/s.
Both have the same momentum.
In grandpa's time automobiles were previously manufactured to be as rigid as possible, whereas autos are now designed to crumple upon impact. Why?
Crumpling on impact extends the amount of time that it will take the vehicle to come to a complete stop. While the impact is the same, this extension of time limits the force.
Law of Conservation of Energy
Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes.
Consider an imaginary miracle car that has a 100% efficient internal combustion enegine and burns fuel that has an energy content of 40 megajoules per liter. If the air resistance and overall fictional forces on the car traveling at highway speed are 500 N, show that the distance the car could travel per liter at this speed is 80 km/L.
From the definition that work = force x distance, simply rearrangement gives distance = work/force. If all 40 million J of energy in 1 L were used to do the work of overoming the air resistance and frictional forces, the distance would be: Distance = work ÷ force = 40,000,000 ÷ 500 N = 80,000 m/L = 80 km/L The important point here is that, even with a hypothetically perfect machine, there is an upper limit of fuel economy dictated by the conservation of energy.
Impulse-Momentum Relationship
Impulse is equal to the change in the momentum of an object that the impulse acts upon: Ft = ∆(mv)
Momentum
Inertia in motion. Momentum = mass x velocity
Why bother using a machine if it cannot multiply work input to achieve greater work output?
It can make work easier, because you do not need to use as much power.
When the velocity of an object is doubled, by what factor is its momentum changed? By what factor is its kinetic energy changed?
Momentum is doubled. Kinetic energy is quadrupled.
When you are driving at 90 km/h, how much more distance do you need to stop than if you were driving at 30 km/h?
Nine times as much distance. The car has nine times as much kinetic energy when it travels three times as fast: 1/2m(3v)^2 = 1/2m9v^2 = 9(1/2mv^2). The friction force is ordinarily the same in either case; therefore, nine times as much work requires nine times as much distance.
Bronco dives from a hovering helicopter and finds his momentum increasing. Does this violate conservation of momentum? Explain.
No, because an external force (gravity) is acting on him to increase his momentum.
If your friend pushes a stroller four times as far as you do while exerting only half the force, which one of you does more work? How much more?
She does more. Work = force x distance. So if I'm pushing it 1 m with a force of 10 N, I'm using 10 J of work. If she pushes it 4 m with a force of 5 meters, she's doing 20 J of work. So she does twice as much work.
Kinetic Energy
The energy of motion, quantified by the relationship: kinetic energy = 1/2mv^2 AKA Net Force x Distance
Potential Energy
The energy that matter possesses due to its position. Gravitational PE = weight x height
For the same force, which cannon imparts a greater impulse to a cannonball: a long cannon or a short one?
The long cannon imparts a greater impulse because the force acts over a longer time.
Impulse
The product of the force acting on an object and the time during which it acts: Impulse = ft
A pair of skater initially at rest push against each other so that they move in opposite directions. What is the total momentum of the two skates as they move apart? Is there a different answer if their masses are not the same?
The total (net) momentum is zero. That's true whatever their masses because each will have the same amount of momentum but in the opposite direction from the other.
In the absence of air resistance, a snowball thrown vertically upward with a certain initial KE returns to its original level with the same KE. When air resistance is a factor affecting the snowball, does it return to its original level with the same, less, or more KE? Does your answer contradict the law of energy conservation?
When air resistance is a factor, the snowball returns with less speed. It therefore has less KE. You can see this directly from the fact that the snowball loses mechanical energy to the air molecules it encounters, so when it returns to its starting point and to its original PE, it has less KE. This does not contradict the law of energy conservation because energy is dissipated from the moving-snowball system, not destroyed.
Newton's second law states that if no net force is exerted on a system, no acceleration occurs. Does it follow that no change in momentum occurs?
Yes, because no acceleration means that no change occurs in velocity or in momentum (mass x velocity). Another line of reasoning is simply that no net force means there is no net impulse and thus no change in momentum.
Newton's third law states that the force a cannon exerts on a cannonball is equal and opposite to the force the cannonball exerts on the cannon. Does it follow that the impulse the cannon exerts on the cannonball is equal and opposite to the impulse the cannonball exerts on the cannon?
Yes, because the interaction between both occurs during the same time interval. Because time is equal and the forces are equal and opposite, the impulses, Ft, are also equal and opposite. Impulse is a vector quantity and can be canceled.
Elastic Collision
A collision in which colliding objects rebound without lasting deformation or the generation of heat.
Inelastic Collision
A collision in which the colliding objects become distorted, generate heat, and possibly stick together.
Machine
A device, such as a lever or pulley, that increases (or decreases) a force or simply changes the direction of a force.
Freddy frog drops vertically from a tree onto a horizontally moving skateboard. The skateboard slows. Give two reasons for the slowing, one in terms of a horizontal friction force between Freddy's feet and the skateboard, and one in terms of momentum conservation.
In terms of force, Freddy's feet are brought up to speed when they make contact with the moving board. The friction force that brings him up to speed is countered by the same amount of force on the board in the opposite direction, slowing the board. In terms of momentum conservation, since no external forces act in the horizontal direction, the momentum after the board catches Freddy equals the momentum before. Since Freddy's mass is added, velocity must decrease.
Does a moving object have impulse?
No. Impulse is not something an object HAS, like momentum. Impulse is what an object can PROVIDE or what it can experience when it interacts with another object. An object cannot posses impulse, just as it cannot posses force.
If a boxer increases the duration of impact to three times as long by riding with the punch, by how much is the force of impact reduced?
The force of impact is only a third of what it would have been if he hadn't pulled back.
If the boxer moves into a punch to decrease the duration of impact by half, by how much is the force of impact increased?
The force of the ipact is twice what it would have been if he had held his head still.
When the mass of a moving object is doubled with no change in speed, by what factor is its momentum changed? B what factor is its kinetic energy changed?
The momentum is doubled, because momentum = mass x velocity. The kinetic energy is also doubled, because KE = 1/2mv^2.
Two lumps of clay with equal and opposite momenta have a head-on collision and come to rest. Is momentum conserved? Is kinetic energy conserved? Why are your answers the same or different?
The net momentum before the lumps collide is zero, and it is still zero after the collision. Momentum is indeed conserved. KE is zero after the collision, but it was greater than zero before the collision. The lumps are warmer after colliding because the initial KE of the lumps is transformed into thermal energy. Momentum only has one form. There is no way to "transform" momentum from one form to another, so it is conserved. But energy comes in various forms and can easily be transformed. No single form of energy such as KE need be conserved.
Work-Energy Theorem
The net work done on an object equals the change in kinetic energy of the object: Work = ∆KE
Efficiency
The percentage of work put into a machine that is converted into useful work output: Efficiency = Useful Energy Output ÷ Total Energy Input
If a golf ball and a Ping-Pong ball move with the same KE, can you say which has the greater speed? Explain in terms of the definition of KE. Similarly, in a gaseous mixture of heavy molecules and light molecules with the same average KE, can you say which have the greater speed?
The ping pong ball and the lighter molecules will have more speed because KE is defined by 1/2mv^2.
Work
The product of the force and the distance moved by the force: W = fd
Energy
The property of a system that enables it to do work
Power
The rate of doing work (or the rate at which energy is expended). Power = work ÷ time
A lunar vehicle is tested on Earth at a speed of 12 km/h. When it travels at the same speed on the Moon, is its momentum greater, less, or the same?
The same
Conservation of Energy for Machines
The work output of any machine cannot exceed the work input. In an ideal machine, where no energy is transformed into thermal energy, work input = work output and (fd) input = (fd) output
A boxer being hit with a punch contrives to extend time for best results, whereas a karate experts delivers a force in a short time for the best results. Why isn't there a contradiction here?
There is no contradiction because the best results for each are quite different. The best results for the boxer is reduced fore, accomplished by maximizing time, and the best results for the karate expert is increased force delivered in minimum time.
On a playground slide, a child has potential energy that decreases by 1000 J while her kinetic energy increases by 900 J. What other form of energy is involved, and how much?
Thermal energy, 100 J.
Two people who weigh the same climb a flight of stairs. The first person climbs the stairs in 30 s, and the second person climbs them in 40 s. Which person does more work? Which uses more power?
They do the same amount of work. The second person uses more power.
Does a moving object have momentum?
Yes, but, like velocity, in a relative sense - that is, with respect to a frame of reference, usually Earth's surface. The momentum possessed by a moving object with respect to a stationary point on Earth may be quite different from the momentum it possesses with respect to another moving planet.