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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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Potential Energy
Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It's like your stored energy that waits to be transformed into another type when the object moves. For example:
  • When you stand on a table, you have gravitational potential energy because of your height above the ground.
  • This energy is directly proportional to both the mass of the object and the height from the ground.
The formula used to calculate gravitational potential energy is:\[PE = mgh\]Where:
  • \( m \) is the mass,
  • \( g \) is the acceleration due to gravity (approximately \( 9.8 \ m/s^2 \) on Earth),
  • \( h \) is the height above the ground.
As you jump off the table, this potential energy is converted into kinetic energy as you move down.
Kinetic Energy
Kinetic energy is the energy an object has due to its motion. It explains why things happen when objects move. For example:
  • When a car drives down the street, it has kinetic energy because it is in motion.
  • Similarly, when you jump off a table, as you fall, your speed increases, hence your kinetic energy increases.
The formula for kinetic energy is:\[KE = \frac{1}{2}mv^2\]Where:
  • \( m \) is the mass, and
  • \( v \) is the velocity of the object.
In scenarios like collisions, such as a car hitting a tree, this kinetic energy is transformed into other forms of energy due to non-conservative forces, contributing to the change in motion and energy loss.
Non-conservative Forces
Non-conservative forces, such as friction, air resistance, and applied forces, are forces that cause energy to change forms or dissipate from a system. These forces do not conserve mechanical energy in a system because they transform mechanical energy into other forms like heat, sound, or internal energy. Examples include:
  • Air resistance when you jump off a table, which reduces the amount of mechanical energy by converting it into heat and sound.
  • Friction between a car's tires and the road, which affects the car's speed and thus its mechanical energy.
  • The impact force when a car hits a tree, turning kinetic energy into deformation energy, sound, and heat.
Whenever these forces act, they prevent the total mechanical energy (sum of potential and kinetic energy) from remaining constant. Instead, energy is transferred to the surroundings or transformed into non-mechanical forms.

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Most popular questions from this chapter

a) If the gravitational potential energy of a 40.0 -kg rock is 500 . J relative to a value of zero on the ground, how high is the rock above the ground? b) If the rock were lifted to twice its original height, how would the value of its gravitational potential energy change?

A \(1.50 \cdot 10^{3}-\mathrm{kg}\) car travels \(2.50 \mathrm{~km}\) up an incline at constant velocity. The incline has an angle of \(3.00^{\circ}\) with respect to the horizontal. What is the change in the car's potential energy? What is the net work done on the car?

A 1.00 -kg block initially at rest at the top of a 4.00 -m incline with a slope of \(45.0^{\circ}\) begins to slide down the incline. The upper half of the incline is frictionless, while the lower half is rough, with a coefficient of kinetic friction \(\mu_{\mathrm{k}}=0.300\). a) How fast is the block moving midway along the incline, before entering the rough section? b) How fast is the block moving at the bottom of the incline?

In 1896 in Waco, Texas, William George Crush, owner of the K-T (or "Katy") Railroad, parked two locomotives at opposite ends of a 6.4 -km-long track, fired them up, tied their throttles open, and then allowed them to crash head- on at full speed in front of 30,000 spectators. Hundreds of people were hurt by flying debris; several were killed. Assuming that each locomotive weighed \(1.2 \cdot 10^{6} \mathrm{~N}\) and its acceleration along the track was a constant \(0.26 \mathrm{~m} / \mathrm{s}^{2},\) what was the total kinetic energy of the two locomotives just before the collision?

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?
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