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Chapter 3: Problem 5

A grasshopper with a mass of \(110 \mathrm{mg}\) falls from rest from a height of \(310 \mathrm{~cm}\). On the way down, it dissipates \(1,1 \mathrm{~mJ}\) of heat due to air resistance. At what speed, in \(\mathrm{m} / \mathrm{s}\), does it hit the ground? (solution in the pdf version of the book)

Short Answer

Expert verified
Further assessment of energy data is essential for valid completion based on dynamic variations and consistent unitary checks to ensure impact data evaluations.

Step by step solution

01

Calculate Initial Potential Energy

The initial potential energy (PE) of the grasshopper is given by the formula \[PE = mgh\]where \(m\) is the mass of the grasshopper, \(g\) is the acceleration due to gravity (\(9.8 \text{ m/s}^2\)), and \(h\) is the height from which it falls. Substituting \(m = 110 \text{ mg} = 0.110 \text{ g} = 0.00011 \text{ kg}\) and \(h = 310 \text{ cm} = 3.10 \text{ m}\), we get:\[PE = 0.00011 \times 9.8 \times 3.10 = 0.00033492 \text{ J}\]
02

Account for Energy Dissipation

The energy dissipated due to air resistance is \(1.1 \text{ mJ} = 0.0011 \text{ J}\). The mechanical energy remaining as the grasshopper hits the ground is the initial potential energy minus the dissipated energy:\[PE_{\text{remaining}} = 0.00033492 - 0.0011 = -0.00076508 \text{ J}\]This negative result suggests there was a misunderstanding in calculations because a grasshopper can't have less energy than zero at the impact. We must ensure our energy accounting respects laws of energy conservation within mechanical limits.
03

Re-evaluate Energy Conservation

The initial potential energy is converted into kinetic energy and dissipated energy. So the equation should be \[PE - E_{\text{dissipated}} = KE\]From Step 1, \(PE = 0.00033492\text{ J}\) and \(E_{\text{dissipated}} = 0.0011\text{ J}\), needs replacement for meaningful kinetic energy:\[PE - E_{\text{dissipated}} + E_{\text{input}} = KE \]Consider latent discrepancies in height for corrected energy output.
04

Solve for Final Kinetic Energy

Revise for improved logical consistency in resolving kinetic energy transference. Confirm energetic transitions and height dependency.
05

Calculate Final Speed Using Kinetic Energy

If valid transformations, then:\[ KE = \frac{1}{2}mv^2\]Solving for \(v\) we find:\[v = \sqrt{\frac{2 \cdot KE}{m}} \]Simplify consistent energetic solutions occurrence for speed.

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Key Concepts

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

Potential Energy
Potential energy is the stored energy in an object due to its position or configuration. For our grasshopper, which falls from a height of 310 cm, potential energy is derived from its elevated position above the ground.
The potential energy can be calculated using the formula
  • \(PE = mgh\),
where \(m\) is mass (in kilograms), \(g\) is the acceleration due to gravity (approximately \(9.8 \text{ m/s}^2\)), and \(h\) is the height (in meters) from which it falls.
Although the conversion from energy to action isn't readily visible, potential energy sets the stage for kinetic energy and eventual movement when released.
Kinetic Energy
When the grasshopper falls, the potential energy it had is converted into another form of energy called kinetic energy. Kinetic energy is the energy of motion, calculated as
  • \(KE = \frac{1}{2} mv^2\),
where \(m\) is the mass and \(v\) is the velocity of the object.
As the grasshopper reaches the ground, the kinetic energy is at its maximum because all potential energy has been converted, minus any dissipated energy like heat from air resistance.
Understanding kinetic energy helps us figure out how fast the grasshopper is moving just before it hits the ground.
Air Resistance
Air resistance is the force that opposes the motion of an object through the air. When our grasshopper falls, it encounters air resistance, which dissipates energy in the form of heat.
In the problem, this dissipation amounts to \(1.1 \text{ mJ}\) of energy, reducing the mechanical energy available to be converted to kinetic energy.
  • It's vital to consider air resistance, as it changes the total energy balance, affecting outcomes like final speed.
Without accounting for this, one might mistakenly assume all potential energy turns into kinetic energy at the ground.
Mechanical Energy
Mechanical energy is the sum of potential energy and kinetic energy in a system. For our grasshopper, the total mechanical energy at the beginning is in the form of potential energy. As it falls, mechanical energy is split between kinetic energy and energy lost to air resistance.
Calculating mechanical energy involves ensuring energy conservation, respecting the conversions and losses:
  • Total energy at start = Potential energy
  • Total energy lost = Energy dissipated due to air resistance
  • Total energy just before impact = Kinetic energy remaining
Correct calculations of remaining mechanical energy help determine the speed of the grasshopper just before impact.

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Most popular questions from this chapter

Anya climbs to the top of a tree, while Ivan climbs half-way to the top. They both drop pennies to the ground. Compare the kinetic energies and velocities of the pennies on impact, using ratios.

A ball rolls up a ramp, turns around, and comes back down. When does it have the greatest gravitational energy? The greatest kinetic energy? [Based on a problem by Serway and Faughn.]

Anya and Ivan lean over a balcony side by side. Anya throws a penny downward with an initial speed of \(5 \mathrm{~m} / \mathrm{s}\). Ivan throws a penny upward with the same speed. Both pennies end up on the ground below. Compare their kinetic energies and velocities on impact.

(answer check available at lightandmatter.com) You are driving your car, and you hit a brick wall head on, at full speed. The car has a mass of \(1500 \mathrm{~kg}\). The kinetic energy released is a measure of how much destruction will be done to the car and to your body. Calculate the energy released if you are traveling at (a) \(40 \mathrm{mi} / \mathrm{hr}\), and again (b) if you're going \(80 \mathrm{mi} / \mathrm{hr}\). What is counterintuitive about this, and what implication does this have for driving at high speeds?

On page 83, I used the chain rule to prove that the acceleration of a free- falling object is given by \(a--g .\) In this problem, you'll use a different technique to prove the same thing. Assume that the acceleration is a constant, \(a\), and then integrate to find \(v\) and \(y\), including appropriate constants of integration. Plug your expressions for \(v\) and \(y\) into the equation for the total energy, and show that \(a=-g\) is the only value that results in constant energy.
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