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# do all five questions Question 1 20 pts Ignoring the effects of air resistance, if a...

do all five questions

Question 1

20 pts

Ignoring the effects of air resistance, if a ball falls freely toward the ground, its total mechanical energy

increases

remains the same

not enough information

decreases

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Question 2

20 pts

A child jumps off a wall from an initial height of 16.4 m and lands on a trampoline. Before the child springs back up into the air the trampoline compresses 1.8 meters. The spring constant of the elastic in the trampoline is K = 288 N/m. Find the mass of the child. Report your answer to 1 decimal place. Please show your work to earn full credit.
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Question 3

20 pts

A 16.0 kg cart is moving with a speed of 14.0 m/s. To what height up a frictionless hill will the cart roll before starting to roll back down? Report your answer to 1 decimal place. Please show your work to earn full credit.
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Question 4

20 pts

An object is thrown upward into the air and follows a parabolic trajectory. As it moves through the air its potential energy

increases, then decreases

stays the same

decreases, then increases

increases only

decreases only

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Question 5

20 pts

An object is thrown upward into the air and follows a parabolic trajectory. As it moves through the air its kinetic energy

decreases, then increases

stays the same

increases only

decreases only

increases, then decreases

Conservation of Energy
Previously, we have looked at how the energy of an object can be changed by work that is being done to it. In this lesson we will be looking at a special case of the Work-Energy Theorem.

https://en.wikipedia.org/wiki/Thermodynamics
Closed Systems
Work is done when a force acts on an object over a distance. An applied force from a person or a friction force is usually easy to identify, but there can be other forces acting on the object at the same time that don't affect the particular motion we are interested in. For instance, a car may be moving on the interstate. We are interested in the force of friction and the force created by the engine to get the car moving. The force of gravity is also acting on the car, but since the car is not dropping into the ground, no work is done. This goes back to the idea that W=Fxd. Since there is no change in distance from the ground W=0J.
However, what if we cannot identify any external forces that are acting on the object? If this is the case, then we have what is known as a closed system. From an energy perspective, a closed system means that nothing is adding or removing energy from our object. If this is true, then the amount of work being done is equal to zero, and the total amount of mechanical energy will stay constant. The picture above represents the closed system using the dotted line to separate the system from the surroundings.

Conservation of Energy
In closed systems energy is conserved. This means no energy is added or removed from the system. In the pendulum system at left the potential and kinetic energy is constantly changing, but overall the total amount of energy is the same at every point in the system. We can set up problems like this using LOL charts.
This means that if our kinetic energy increases during the problem, the potential energy must decrease. If the potential energy increases, the kinetic energy must decrease. Again, we are not adding or taking away energy from the object, we are just changing its form.

Another common example of the idea of conservation of momentum is the roller coaster. Some work must be done to get the object to the top of the first hill. If we ignore friction, conservation of energy is observed. This is why the roller coaster will never be able to reach a higher position than where it started. It also means that the coaster will be going the fastest through their lowest positions.
Example Problem 1
A roller coaster is traveling 40 m/s when it reaches the bottom of the first hill. How tall must the hill be in order to reach this velocity?

GPE = KE
LOL Graph
The first step in solving this problem is to determine the starting and ending situations. The start would be the top of the hill where the coaster has all GPE. The ending position is at the bottom of the hill where the coaster has all KE. Ignoring friction, the LOL graph will look like this. Since this is a closed system, we place the roller coaster in the circle but there are no arrows coming in or out of the circle.
Based on the LOL chart we can write the following equations:
GPE = KE
m * g * h = .5 * m * v2
A quick look at this equation tells us that the mass variable is found on both sides of this equation. This means we can cancel the m's and rewrite the equation as:
g * h = .5 * v2
Inserting our numbers we get:
9.8 * h = .5 * (40)2
h = 81.6 m tall
It is important to notice that gravity is acting on both objects. In fact, it is the force of gravity that is responsible for the motion of both the roller coaster. Gravity is not an external force. It is always present, and is acting on the object because of it's own mass. So an object can still experience conservation of energy if gravity is the force causing the motion or the transfer of energy.
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Example Problem 2
Starting from rest, a 25 kg child slides down a frictionless slide. The slide is 3.0 m high. What is the child's speed at the bottom of the slide?

GPE = KE
LOL Graph
In this situation the child starts at the top of the slide which means all GPE. At the bottom of the slide the child has all KE. Notice, the LOL chart looks similar to the one used in Example 1.
Using the LOL chart we know that GPE = KE so:
m * g * h = .5 * m * v2
Canceling the mass gives us:
g * h = .5 * v2
9.8 * 3 = .5 * v2
v = 7.67 m/s
The child's speed at the bottom of the slide is 7.67 m/s.
Example Problem 3
A 755 N diver dives from a board 10.0 m above the water's surface. If the diver leaves the board with an initial speed of 2.00 m/s, what is the diver's speed when striking the water?

KE + GPE = KE
LOL Graph
The diver has both PE and KE when leaving the board, so the left side of the LOL chart will have both. Once the diver hits the water, there is no more PE. It has all been converted to KE. So the right side of the equation will only have KE.
Using the LOL chart we know that KE + GPE = KE so:
(.5 * m * v2) + (m * g * h) = .5 * m * v2
Since mass is included in each term of the equation we can still cancel the mass. This gives us:
(.5 * v2) + (g * h) = .5 * v2
(.5 * 22) + (9.8 * 10) = .5 v2
2 + 98 = .5 v2
100/.5 = v2
200 = v2
v = 14.1 m/s
The diver's speed at the end of the dive is 14.1 m/s.
Example Problem 4
A ball is released from an initial height of 10 m so that as the ball stops it compresses a spring with 40 cm. The spring constant is K = 50 N/m. Find the mass of the ball.

GPE = EPE
LOL Graph
Initially, the ball is held above the spring having only GPE. As it falls, the GPE becomes KE, but the final situation here is after the ball has stopped. The energy transfer

Therefore, as God’s chosen people, holy and dearly loved, clothe yourselves with compassion, kindness, humility, gentleness and patience. Col 3:12

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