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Chapter 6: Problem 11

a) If you jump off a table onto the floor, is your mechanical energy conserved? If not, where does it go? b) A car moving down the road smashes into a tree. Is the mechanical energy of the car conserved? If not, where does it go?

Short Answer

Expert verified
a) A person jumping off a table onto the floor. b) A car moving down the road and smashing into a tree. Answer: a) Mechanical energy is not conserved in this scenario. As the person falls, energy is lost to air resistance, sound energy upon impact, and internal energy transferred to the tissues of the body. b) Mechanical energy is also not conserved in this case. The car's kinetic energy is transformed into other forms of energy such as internal energy of the car's deformation, heat, and energy transferred to the tree and the ground (in the form of vibrations and sounds).

Step by step solution

01

Understand the conservation of mechanical energy

The principle of conservation of mechanical energy states that the total mechanical energy of an isolated system remains constant if no non-conservative forces, such as friction or air resistance, are acting on the system.
02

Analyze the situation of the jump

Initially, when the person is on the table, they have gravitational potential energy. As they jump and fall to the floor, this potential energy is converted into kinetic energy.
03

Determine if mechanical energy is conserved

As the person falls, air resistance acts as a non-conservative force. When they reach the ground, energy will be lost to the surroundings in the form of sound energy (the sound produced upon impact) and internal energy (energy transferred to the tissues of the body upon impact). Thus, the mechanical energy is not conserved in this scenario. For scenario (b):
04

Understand the conservation of mechanical energy

As mentioned earlier, the principle of conservation of mechanical energy states that the total mechanical energy of an isolated system remains constant if no non-conservative forces are acting on the system.
05

Analyze the car's motion and impact

Before the collision, the car has kinetic energy due to its motion. Upon smashing into the tree, the car comes to a stop.
06

Determine if mechanical energy is conserved

The mechanical energy of the car is not conserved in this case, as non-conservative forces act on it during the collision. These forces include friction between the car and the road, the force exerted by the tree on the car, and internal forces within the car itself. The kinetic energy of the car is transformed into other forms of energy, such as the internal energy of the car's deformation and heat, and energy transferred to the tree and the ground (vibrations and sounds).

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Most popular questions from this chapter

A ball is thrown up in the air, reaching a height of \(5.00 \mathrm{~m}\). Using energy conservation considerations, determine its initial speed.

A mass of \(1.00 \mathrm{~kg}\) attached to a spring with a spring constant of \(100 .\) N/m oscillates horizontally on a smooth frictionless table with an amplitude of \(0.500 \mathrm{~m} .\) When the mass is \(0.250 \mathrm{~m}\) away from equilibrium, determine: a) its total mechanical energy; b) the system's potential energy and the mass's kinetic energy; c) the mass's kinetic energy when it is at the equilibrium point. d) Suppose there was friction between the mass and the table so that the amplitude was cut in half after some time. By what factor has the mass's maximum kinetic energy changed? e) By what factor has the maximum potential energy changed?

A body of mass \(m\) moves in one dimension under the influence of a force, \(F(x)\), which depends only on the body's position. a) Prove that Newton's Second Law and the law of conservation of energy for this body are exactly equivalent. b) Explain, then, why the law of conservation of energy is considered to be of greater significance than Newton's Second Law.

A spring with \(k=10.0 \mathrm{~N} / \mathrm{cm}\) is initially stretched \(1.00 \mathrm{~cm}\) from its equilibrium length. a) How much more energy is needed to further stretch the spring to \(5.00 \mathrm{~cm}\) beyond its equilibrium length? b) From this new position, how much energy is needed to compress the spring to \(5.00 \mathrm{~cm}\) shorter than its equilibrium position?

A high jumper approaches the bar at \(9.0 \mathrm{~m} / \mathrm{s}\). What is the highest altitude the jumper can reach, if he does not use any additional push off the ground and is moving at \(7.0 \mathrm{~m} / \mathrm{s}\) as he goes over the bar?
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