In Sacramento a SUV is traveling south on Truxel crashes into an empty school bus, traveling east on San Juan. The collision is perfectly inelastic. Find the velocity of the wreck just after collision
Find the direction in which the wreck initially moves
A ball is moving in the positive direction when it is struck dead center by a ball moving in the positive direction. After collision the ball moves at degrees from the positive axis.
To significant digit accuracy fill out the following table:
ball ball ball ball
Momentum before
Momentum after collision
Find the velocity and direction of the ball.
Use the table to prove momentum is conserved.
Prove that kinetic energy is not conserved.
Students are doing an experiment on the lab table. A steel ball is rolled down a small ramp and allowed to hit the floor. Its impact point is carefully marked. Next a second ball of the same mass is put upon a set screw and a collision takes place such that both balls go off at an angle and hit the floor. All measurements are taken with a meter stick on the floor with a co-ordinate system such that just below the impact point is the origin. The following data is collected: no collision:
target ball: in the direction of motion and perpendicular to the direction of motion
From this data predict the impact position of the other ball.
One of the lab groups declares that the data on the floor alone demonstrate to a % accuracy that the collision was elastic. Show their reasoning.
Another lab group says they can’t make that determination without knowing the velocity the balls have on impact. They ask for a timer. The instructor says you don’t need one; use your meter stick. Explain.
Design an experiment to prove momentum conservation with balls of different masses, giving apparatus, procedure and design. Give some sample numbers.
Answers to Selected Problems
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c.
d.
e. due to Newton’s third law
to the left
a. b.
a. b.
c. yes,
to the left
a. to the left b. tree experienced same average force of but to the right
c. .
d. about “g”s of acceleration
a. no change b. the last two cars
a.
a. b.
a. b. of
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Chapter 9: Energy and Force Version 2
The Big Idea
The law of conservation of momentum states that in any closed system (including the universe) the total quantity of momentum is constant. Momentum can be transferred from one body to another, but none is lost or gained. If a system has its momentum changed from the outside it is caused by an impulse, which transfers momentum from one body to another.
When any two bodies in the universe interact, they can exchange energy, momentum, or both. We saw in an earlier chapter that the law of conservation of energy states that in any closed system (including the universe) the total quantity of energy remains fixed. Energy is transferred from one form to another, but not lost or gained. If energy is put into a system from the outside or vice versa it is often in the form of work, which is a transfer of energy between bodies.
At this point, we have an opportunity to explore the relationship between force, energy and work. These are really important concepts in physics, so we should take our time to understand them. At this point the situation can be summarized like this: energy is the capacity to create motion or change, force is what creates the change, and work is a book keeping tool to keep track of forces.
Math of Force, Energy, and Work
When an object moves in the direction of an applied force, we say that the force does work on the object. Note that the force may be slowing the object down, speeding it up, maintaining its velocity --- any number of things. In all cases, the net work done is given by this formula: In other words, if an object has traveled a distance under force , the work done on it will equal to multiplied by the component of along the object's path. Consider the following example of a block moving horizontally with a force applied at some angle:
Here the net work done on the object by the force will be .
Work-Energy Principle
The reason the concept of work is so useful is because of a theorem, called the work-energy principle, which states that the change in an object's kinetic energy is equal to the net work done on it: Although we cannot derive this principle in general, we can do it for the case that interests us most: constant acceleration. In the following derivation, we assume that the force is along motion. This doesn't reduce the generality of the result, but makes the derivation more tractable because we don't need to worry about vectors or angles.
Recall that an object's kinetic energy is given by the formula: Consider an object of mass accelerated from a velocity to under a constant force. The change in kinetic energy, according to [2], is equal to: Now let's see how much work this took. To find this, we need to find the distance such an object will travel under these conditions. We can do this by using the third of our 'Big three' equations, namely: Plugging in [6] and Newton's Third Law, , into [2], we find: which was our result in [4].
Using the Work-Energy Principle
The Work-Energy Principle can be used to derive a variety of useful results. Consider, for instance, an object dropped a height under the influence of gravity. This object will experience constant acceleration. Therefore, we can again use equation [6], substituting gravity for acceleration and for distance: In other words, the work performed on the object by gravity in this case is . We refer to this quantity as gravitational potential energy; here, we have derived it as a function of height. For most forces (exceptions are friction, air resistance, and other forces that convert energy into heat), potential energy can be understood as the ability to perform work.
Spring Force
A spring with spring constant a distance from equilibrium experiences a restorative force equal to: This is a force that can change an object's kinetic energy, and therefore do work. So, it has a potential energy associated with it as well. This quantity is given by: The derivation of [10] is left to the reader. Hint: find the average force an object experiences while moving from to while attached to a spring. The net work is then this force times the displacement. Since this quantity (work) must equal to the change in the object's kinetic energy, it is also equal to the potential energy of the spring. This derivation is very similar to the derivation of the kinematics equations --- look those up.
Summary of Key Equations and Definitions
Here is a summary of important concepts from this and the past few chapters. The point of this chapter is to combine all of our knowledge so far to solve new kinds of problems.
One important type of problem is called a collision problem. In cases where collisions are elastic, kinetic energy and momentum are conserved. In inelastic collisions, only momentum is conserved.
Key Concepts
An impulse occurs when momentum is transferred from one system to another. You can always determine the impulse by finding the changes in momentum, which are done by forces acting over a period of time. If you graph force vs. time of impact, the area under the curve is the impulse.
Work is simply how much energy was transferred from one system to another system. You can always find the work done on an object (or done by an object) by determining how much energy has been transferred into or out of the object through forces. If you graph force vs. distance, the area under the curve is work. (The semantics take some getting used to: if you do work on me, then you have lost energy, and I have gained energy.)
Key Applications
When working a problem that asks for height or speed, energy conservation is almost always the eas
iest approach.
Potential energy of gravity, , is always measured with respect to some arbitrary ‘zero’ height defined to be where the gravitational potential energy is zero. You can set this height equal to zero at any altitude you like. Be consistent with your choice throughout the problem. Often it is easiest to set it to zero at the lowest point in the problem.
Some problems require you to use both energy conservation and momentum conservation. Remember, in every collision, momentum is conserved. Kinetic energy, on the other hand, is not always conserved, since some kinetic energy may be lost to heat.
If a system involves no energy losses due to heat or sound, no change in potential energy and no work is done by anybody to anybody else, then kinetic energy is conserved. Collisions where this occurs are called elastic. In elastic collisions, both kinetic energy and momentum are conserved. In inelastic collisions kinetic energy is not conserved; only momentum is conserved.
Sometimes energy is “lost” when crushing an object. For instance, if you throw silly putty against a wall, much of the energy goes into flattening the silly putty (changing intermolecular bonds). Treat this as lost energy, similar to sound, chemical changes, or heat. In an inelastic collision, things stick, energy is lost, and so kinetic energy is not conserved.
When calculating work, use the component of the force that is in the same direction as the motion. Components of force perpendicular to the direction of the motion don’t do work. (Note that centripetal forces never do work, since they are always perpendicular to the direction of motion.)
When calculating impulse the time to use is when the force is in contact with the body.
Work and Energy Examples
Example 1
Question: A pile driver lifts a 500 kg mass a vertical distance of 20 m in 1.1 sec. It uses 225 kW of supplied power to do this.
a) How much work was done by the pile driver?
b) How much power was used in actually lifting the mass?
c) What is the efficiency of the machine? (This is the ratio of power used to power supplied.)
d) The mass is dropped on a pile and falls 20 m. If it loses 40,000 J on the way down to the ground due to air resistance, what is its speed when it hits the pile?
Answer:
a) We will use the equation for work and plug in the known values to get the amount of work done by the pile driver.
b) We will use the power equation and plug in the known values to the power used.
c) This is simply a division problem.
d) We have already solved for the amount of energy the mass has after the pile driver performs work on it (it has ). If on the way down it loses due to air resistance, then it effectively has of energy. So we will set the kinetic energy equation equal to the total energy and solve for v. This will give us the velocity of the mass when it hits the ground because right before the mass hits the ground, all of the potential energy will have been converted into kinetic energy.
Energy and Force Problem Set
At 8:00 AM, a bomb exploded in mid-air. Which of the following is true of the pieces of the bomb after explosion? (Select all that apply.) The vector sum of the momenta of all the pieces is zero.
The total kinetic energy of all the pieces is zero.
The chemical potential energy of the bomb has been converted entirely to the kinetic energy of the pieces.
Energy is lost from the system to sound, heat, and a pressure wave.
A rock with mass is dropped from a cliff of height . What is its speed when it gets to the bottom of the cliff?
None of the above
Two cats, Felix and Meekwad, collide. Felix has a mass of and an initial velocity of to the west. Meekwad has a mass of and is initially at rest. After the collision, Felix has a velocity of to the west and Meekwad has a velocity of to the west. Verify that momentum was conserved. Then, determine the kinetic energies of the system before and after the collision. What happened?! (All numbers are exact.)
You are at rest on your bicycle at the top of a hill that is tall. You start rolling down the hill. At the bottom of the hill you have a speed of . Your mass is . Assuming no energy is gained by or lost to any other source, which of the following must be true? The wind must be doing work on you.
You must be doing work on the wind.
No work has been done on either you or the wind.
Not enough information to choose from the first three.
A snowboarder, starting at rest at the top of a mountain, flies down the slope, goes off a jump and crashes through a second-story window at the ski lodge. Retell this story, but describe it using the language of energy. Be sure to describe both how and when the skier gained and lost energy during her journey.
An airplane with mass is traveling with a speed of . What is the kinetic energy of the plane at this speed?
A wind picks up, which causes the plane to lose per second.
How fast is the plane going after seconds?
A roller coaster begins at rest above the ground, as shown. Assume no friction from the wheels and air, and that no energy is lost to heat, sound, and so on. The radius of the loop is . Find the speed of the roller coaster at points , and .
Assume that % of the initial potential energy of the coaster is lost due to heat, sound, and air resistance along its route. How far short of point will the coaster stop?
Does the coaster actually make it through the loop without falling? (Hint: You might review the material from Chapter 6 to answer this part.)
In the picture above, a baby on a skateboard is about to be launched horizontally. The spring constant is and the spring is compressed . For the following questions, ignore the small energy loss due to the friction in the wheels of the skateboard and the rotational energy used up to make the wheels spin. What is the speed of the baby after the spring has reached its uncompressed length?
After being launched, the baby encounters a hill high. Will the baby make it to the top? If so, what is his speed at the top? If not, how high does he make it?
Are you finally convinced that your authors have lost their minds? Look at that picture!
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When the biker is at the top of the ramp shown above, he has a speed of and is at a height of . The bike and person have a total mass of . He speeds into the contraption at the end of the ramp, which slows him to a stop. What is his initial total energy? (Hint: Set at the very bottom of the ramp.)
What is the length of the spring when it is maximally compressed by the biker? (Hint: The spring does not compress all the way to the ground so there is still some gravitational potential energy. It will help to draw some triangles.)
An elevator in an old apartment building in Switzerland has four huge springs at the bottom of the shaft to cushion its fall in case the cable breaks. The springs have an uncompressed height of about meter. Estimate the spring constant necessary to stop this elevator, following these steps: First, guesstimate the mass of the elevator with a few passengers inside.
Now, estimate the height of a five-story building.
Lastly, use conservation of energy to estimate the spring constant.
You are driving your buddy to class in a car of mass at a speed of . You and your passenger each have of mass. Suddenly, a deer runs out in front of your car. The coefficient of friction between the tires and the freeway cement is . In addition there is an average force of friction of exerted by air resistance, friction of the wheels and axles, etc. in the time it takes the car to stop. What is your stopping distance if you skid to a stop?
What is your stopping distance if you roll to a stop (i.e., if the brakes don’t lock)?
You are skiing down a hill. You start at rest at a height above the bottom. The slope has a grade. Assume the total mass of skier and equipment is . Ignore all energy losses due to friction. What is your speed at the bottom?
If, however, you just make it to the bottom with zero speed what would be the average force of friction, including air resistance?
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Two horrific contraptions on frictionless wheels are compressing a spring by compared to its uncompressed (equilibrium) length. Each of the vehicles is stationary and they are connected by a string. The string is cut! Find the speeds of the masses once they lose contact with the spring.
You slide down a hill on top of a big ice block as shown in the diagram. Your speed at the top of the hill is zero. The coefficient of kinetic friction on the slide down the hill is zero . The coefficient of kinetic friction on the level part just beneath the hill is . What is your speed just as you reach the bottom of the hill?
How far will you slide before you come to a stop?
A woman falls from a height of and lands on a springy mattress. If the springs compress by , what is the spring constant of the mattress?
no energy is lost from the system, what height will she bounce back up to?
Marciel is at rest on his skateboard (total mass ) until he catches a ball traveling with a speed of . The baseball has a mass of . What percent of the original kinetic energy is transferred into heat, sound, deformation of the baseball, and other non-mechanical forms when the collision occurs?
CK-12 People's Physics Book Version 2 Page 8